Finite Sample Performance of the Robust Stein Estimator in the Presence of Multicollinearity and Outliers

1. Introduction

Linear regression analysis is an important statistical tool that helps to model a relationship between the response variable ($\mathit{y}$) and the predictors ($\mathit{X}$), and it is often applied in all fields of study. A linear model has the form

$$\mathit{y}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon },$$

(1)

where $\mathit{y}$ is an $n\times 1$ vector of observed values of the dependent variable, $\mathit{X}={[x_1,x_2,\dots ,x_n]}^T$ is an $n\times p$ matrix with a row vector $x_i={[x_{i1},x_{i2},\dots ,x_{ip}]}^{\prime }$ of components $x_{ij}$ of the regressors, $i=1,2,3,\dots ,p$, $\mathit{\beta }$ is a vector $p\times 1$ of unknown parameters and $\epsilon$ is an $n\times 1$ vector of errors. Under classical assumptions that the errors have zero mean, constant variance, and are uncorrelated, then the ordinary least-squares (OLS) estimate $\widehat{\beta }$ is the best linear unbiased estimator (BLUE) of $\beta$ according to the Gauss-Markov theorem. The OLS estimate is BLUE even without assuming normally distributed residuals, and minimizes the sum of squared residuals. If an error term ($\epsilon$) follows a normal $\epsilon \sim \mathcal{N}(0,{\sigma }^{2}I)$ distribution, this assumption of normality is essential for performing hypothesis tests and constructing confidence intervals. If there is more than one predictor variable, then the analysis is referred to as multiple linear regression analysis [2]. A multiple regression model could be used to describe this relationship is

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots +{\beta }_p{x}_p+\epsilon .$$

(2)

Applying the ordinary least squares estimator (OLSE) in simple or multiple linear regression always requires some assumptions, that is, normality of the error terms, equal variance of the error terms, absence of outliers, leverage points, and multicollinearity. Therefore, the OLS estimate is the best method in regression analysis if the assumptions are met. However, if these assumptions are not satisfied, the results can easily be affected [3].

Various problems can occur in the estimation of the model parameters, one of which is if the assumption of normality is not fulfilled, or due to outliers in the data [4]. According to [5,6], the normality of error distributions is based on the central limit theorem, which is a limit theorem based on approximations. Hampel and Huber [5] critically challenged the prevailing assumption that the OLSE remains approximately optimal under conditions of approximate normality. This is because it has been shown that typical error distributions of real-world datasets usually deviate slightly from normality and, in most cases, are tailed. Additionally, outliers in the dependent variable lead to large residual values, which further result in the failure of the normality assumption of the error terms. According to [7], the data that come from the regression line is an outlier. Outliers are data that have different characteristics from other non-outlier data. There exist several statistical methods used to detect outliers, such as the Cook distance index plot and the potential residual plot (see, for example, [8]). The existence of outliers can affect the process of data analysis and errors in making conclusions, but the existence of outliers can be overcome by several methods, which include robust regression [9]. Several robust regression estimators have been provided to address the problem of outliers in different regression models; these include the Huber maximum estimator (HME), the modified maximum likelihood estimator (MME), the least trimmed squares estimator (LTSE), the least median of squares estimator (LMSE), S estimator (SE) [10].

In addition, multicollinearity is another pressing challenge in linear regression models that occurs when predictor variables have a high degree of correlation. This problem causes maximum likelihood estimates to be unstable and inefficient. Therefore, many biased estimators have been provided to address this problem in different regression models [11,12]. However, both multicollinearity and outliers can exist simultaneously in a model. To address both issues, various estimators have been combined for robust estimation in order to handle multicollinearity and outliers [1,13,14]. Recently, the Stein estimator has gained popularity as an alternative to OLSE and performs well in handling correlated regressors. However, it is sensitive to outliers in the y-direction [1]. In this study, we examine a robust version of the Stein estimator, which is the combination of the HME and the James-Stein estimator (JSE), which was originally proposed by [1] and can handle both multicollinearity and outliers simultaneously. This estimator’s performance is assessed and compared with that of other robust regression methods utilizing different simulation scenarios and real data applications.

2. Materials and Methods

In this section, attention will be on the robust methods considered in this study and their properties. The selected estimators were chosen for their theoretical significance as well as their easy implementation and computational viability due to their availability in R. Effective analysis is possible without adding a great deal of computational complexity by using these traditional methods. Because these estimators are available in R, the study is more accessible and reproducible, which makes it useful for other researchers. This decision guarantees an effective combination of computational effectiveness and strong statistical characteristics.

2.1. Review on Robust Estimators

When the data are contaminated by one or a few outliers, the issue of detecting such observations becomes challenging. In general, however, datasets often contain multiple outliers or clusters of influential observations, and they may also exhibit multicollinearity among their variables. These two factors provide additional insight into the co-occurrence of both outliers and multicollinearity, which complicates regression analysis, with biased and unstable parameter estimates as a common result. Alma [15] points out that robust estimation is an important method for analyzing data contaminated by outliers. It is an approach to regression analysis that aims to get past some of the limitations of classical parametric and nonparametric approaches. Well-performing estimates are robust estimates designed to become less sensitive to outliers. Here, we present a review of existing robust estimators used in linear regression.

2.1.1. The Huber Maximum Likelihood Estimator (HME)

The most common general robust method is M-estimates, introduced by [6]. The M in M- estimates stands for maximum likelihood type. The M-estimator minimizes the objective function.

$$\sum _{i=1}^n\rho \left(r_i\right)=\sum _{i=1}^n\rho \left({\displaystyle \frac{e_i\left(\beta \right)}{\sigma }}\right),$$

(3)

where $r_i={\displaystyle \frac{e_i\left(\beta \right)}{\sigma }}$ are called scaled residuals. Differentiating with respect to $\beta$, assuming $\sigma$ is fixed, and setting the partial derivatives to zero, we obtain the normal

equations

$$\sum _{i=1}^n\psi \left({\displaystyle \frac{e_i\left(\beta \right)}{\sigma }}\right)x_i=0.$$

(4)

Let $X=\left[{\mathbf{1}}_n\dot{X}\right]$ denote the $n\times (p+1)$ design matrix, where ${\mathbf{1}}_n$ is the intercept column and $\dot{X}$ contains the p predictors. The ith observation can be written as ${\mathbf{x}}_i^{\prime }=(1,x_{i1},x_{i2},\dots ,x_{ip})$, where $i=1,2,\dots ,n$ (observations) and $j=1,2,\dots ,p$ (predictors). We consider the spectral decomposition $T^{\prime }\left(X^{\prime }X\right)T=\Lambda =diag\left({\lambda }_j\right),j=1,2,\dots ,p^{*},(p^{*}=p+1)$.

Following [1], the model in Eq. (1) is transformed as follows

$$y_i=\sum _{j=1}^{p^{*}}{\alpha }_j{h}_{ij}+{\epsilon }_i,i=1,2,\dots ,n$$

(5)

where ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_{p^{*}}$, and T is a $(p^{*}\times p^{*})$ orthogonal matrix whose columns are eigenvectors corresponding to these eigenvalues. The transformation is defined as $\mathbf{H}=\dot{X}T$, $\alpha =T^{\prime }\beta$, and $T\left(X^{\prime }X\right)T^{\prime }={\mathbf{H}}^{\prime }\mathbf{H}=\Lambda .$ Therefore, the OLS of $\alpha$ is written as follows:

$${\widehat{\alpha }}_{LS}={\Lambda }^{-1}{H}^{\prime }y.$$

(6)

The M-estimator of $\alpha$ is

$${\widehat{\alpha }}_M=min\sum _{i=1}^n\rho \left({\displaystyle \frac{{\epsilon }_i}{\sigma }}\right)=min\sum _{i=1}^n\rho \left({\displaystyle \frac{y_i-{\sum }_{j=1}^{p^{*}}{\alpha }_j{h}_{ij}}{\sigma }}\right).$$

(7)

where $\rho$ is a robust function and $\sigma$ is a scale parameter. The estimator ${\widehat{\alpha }}_M$ is obtained as the solution to the M-estimating equations (3) and (4).

2.1.2. James Stein Estimator (JSE)

As a remedy to the problem of multicollinearity, the James-Stein estimator (JSE), originally proposed by [16] and [17], applies shrinkage to the ordinary least squares estimator to improve estimation efficiency under correlated regressors. In the standard regression context, the JSE is given by

$${\widehat{\mathit{\beta }}}_{\mathrm{JSE}}=\left(1-{\displaystyle \frac{(p-2){\widehat{\sigma }}^2}{{\widehat{\mathit{\beta }}}_{\mathrm{OLS}}^{\prime }{\mathbf{X}}^{\prime }\mathbf{X}{\widehat{\mathit{\beta }}}_{\mathrm{OLS}}}}\right){\widehat{\beta }}_{\mathrm{OLS}},\mathrm{for}p>2,$$

(8)

where ${\widehat{\mathit{\beta }}}_{\mathrm{OLS}}$ is the ordinary least squares estimator and ${\widehat{\sigma }}^2$ is the estimated error variance.

In the transformed $\alpha$-space defined in Eq. (5), following [1], the JSE can be expressed as

$${\widehat{\mathit{\alpha }}}_{\mathrm{JSE}}=c{\widehat{\mathit{\alpha }}}_{\mathrm{LS}},$$

(9)

where the shrinkage factor c is given by

$$c={\displaystyle \frac{{\widehat{\mathit{\alpha }}}_{\mathrm{LS}}^{\prime }{\widehat{\mathit{\alpha }}}_{\mathrm{LS}}}{{\widehat{\mathit{\alpha }}}_{\mathrm{LS}}^{\prime }{\widehat{\mathit{\alpha }}}_{\mathrm{LS}}+{\sigma }^2\mathrm{tr}\left({\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-1}\right)}}=\sum _{j=1}^{p^{*}}{\displaystyle \frac{{\lambda }_j{\widehat{\alpha }}_j^2}{{\sigma }^2+{\lambda }_j{\widehat{\alpha }}_j^2}}.$$

(10)

The equivalence between the two forms in Eq. (10) follows from the spectral decomposition, noting that $\mathrm{tr}\left({\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-1}\right)={\sum }_{j=1}^{p^{*}}{\lambda }_j^{-1}$ and the canonical transformation diagonalizes the covariance structure into independent components along the eigenvector directions.

2.1.3. The Robust Stein Estimator (RSE)

The robust Stein estimator was first introduced by [1] as an alternative method to the Stein and ridge estimators to improve the accuracy of the estimation. However, both methods are non-robust to outliers. In the author’s study, it was indicated that the Stein estimator is sensitive to outliers in the y-direction. Thus, there is a need to propose the robust Stein estimator [1], which is defined as follows

$${\widehat{\alpha }}_{M-JSE}=c^{*}{\widehat{\alpha }}_M,$$

(11)

where, ${\widehat{\alpha }}_M$ is the M-estimate of $\alpha$, The shrinkage factor $c^{*}$ in the Robust Stein (M–Stein) estimator is often defined using the following components

$$c^{*}=\sum _{j=1}^{p^{*}}\left({\displaystyle \frac{{\lambda }_j{\alpha }_M^2}{{\Psi }_{jj}+{\lambda }_j{\alpha }_M^2}}\right),$$

(12)

where,

• ${\lambda }_j$ are the ordered eigenvalues of the design information matrix (e.g., $X^{\top }X$ in linear models),
• ${\alpha }_{M,j}$ are the components of the robust estimator ${\widehat{\mathit{\alpha }}}_M$,
• ${\Psi }_{jj}$ are the diagonal entries of the variance-covariance matrix $Var\left({\widehat{\alpha }}_M\right)$.

The robust covariance matrix $\mathbf{\Psi }$ is estimated using the sandwich estimator [6]

$$\widehat{\mathbf{\Psi }}={\widehat{A}}^{-2}\widehat{B},\mathrm{where}{\widehat{A}}^2={\displaystyle \frac{1}{n-p^{*}}}\sum _{i=1}^n{\left[{\psi }^{\prime }\left({\displaystyle \frac{e_i}{\widehat{\sigma }}}\right)\right]}^2,\widehat{B}={\displaystyle \frac{{\widehat{\sigma }}^2}{{(n-p^{*})}^2}}\sum _{i=1}^n{\left[\psi \left({\displaystyle \frac{e_i}{\widehat{\sigma }}}\right)\right]}^2,$$

(13)

and $\widehat{\sigma }$ is a robust scale estimate. The approximate covariance of ${\widehat{\alpha }}_{\mathrm{M}-\mathrm{JSE}}$ is

$$Cov\left({\widehat{\mathit{\alpha }}}_{\mathrm{M}-\mathrm{JSE}}\right)\approx c^{*}Cov\left({\widehat{\mathit{\alpha }}}_M\right){\left(c^{*}\right)}^{\prime }=c^{*}\Psi {\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-1}{\left(c^{*}\right)}^{\prime }.$$

(14)

The approximate bias is

$$Bias\left({\widehat{\mathit{\alpha }}}_{\mathrm{M}-\mathrm{JSE}}\right)\approx \mathbb{E}\left(c^{*}{\widehat{\mathit{\alpha }}}_M\right)-\alpha =(c^{*}-1)\alpha .$$

(15)

The matrix mean squared error (MMSE) is approximately

$$\mathrm{MMSE}\left({\widehat{\mathit{\alpha }}}_{M-JSE}\right)\approx c^{*}\Psi {\left({\mathbf{X}}^{\prime }\mathbf{X}\right)}^{-1}{\left(c^{*}\right)}^{\prime }+Bias\left({\widehat{\mathit{\alpha }}}_{M-JSE}\right)Bias{\left({\widehat{\alpha }}_{M-JSE}\right)}^{\prime }.$$

(16)

The scalar mean squared error (SMSE) is approximately

$$\mathrm{SMSE}\left({\widehat{\mathit{\alpha }}}_{M-JSE}\right)\approx {\left(c^{*}\right)}^2\sum _{j=1}^{p^{*}}{\displaystyle \frac{{\Psi }_{jj}}{{\lambda }_j}}+{(c^{*}-1)}^2\sum _{j=1}^{p^{*}}{\alpha }_j^2,$$

(17)

which can also be expressed as

$$\mathrm{SMSE}\left({\widehat{\mathit{\alpha }}}_{M-JSE}\right)\approx \sum _{j=1}^{p^{*}}{\displaystyle \frac{{\Psi }_{jj}{\lambda }_j{\alpha }_j^4}{{({\Psi }_{jj}+{\lambda }_j{\alpha }_j^2)}^2}}+\sum _{j=1}^{p^{*}}{\displaystyle \frac{{\Psi }_{jj}^2{\alpha }_j^2}{{({\Psi }_{jj}+{\lambda }_j{\alpha }_j^2)}^2}}.$$

(18)

2.1.4. The Least Median of Squares Estimator (LMSE)

Since M-estimators are not robust to high-leverage outliers, we want to find some methods that can have high BP. [18] defined the least median of squares (LMS) estimator as

$${\widehat{\beta }}_{LMS}=\underset{\beta }{argmin}med\left\{e_i^2\right\},$$

(19)

which replaces the sum by the median in the LSE.

2.1.5. The Least Trimmed Squares Estimator (LTSE)

Another regression estimator that has a breakdown point (BP) of nearly 50% is the least trimmed square (LTS) estimator proposed by [19]. Traditional OLS methods are highly sensitive to outliers, meaning a few extreme data points can dramatically affect the estimated model. LTSE is designed to be more robust to these outliers by focusing on a subset of the data with the smallest residuals. The estimator chooses the regression coefficients $\beta$ to minimize the sum of the smallest h of the squared residuals and is defined as

$${\widehat{\beta }}_{LTS}=\underset{\beta }{argmin}\sum _{i=1}^h{e}_{\left(i\right)}^2\left(\beta \right),$$

(20)

where, $e_{\left(i\right)}^2\left(\beta \right)$ represents the i-th ordered squared residuals $e_{\left(1\right)}^2\le \dots \le e_{\left(n\right)}^2$ and h is called the trimming constant which must satisfy $n/2<h\le n$. The constant h determines the BP of the LTS estimator. Typically, $h=\lfloor n/2\rfloor +\lfloor (p+1)/2\rfloor$ can attain the maximum, where $\lfloor ·\rfloor$ denotes the floor function (rounding down to the next smallest integer). When $h=n$, LTS is exactly equivalent to the LS estimator whose BP is zero.

2.1.6. The S-Estimator (SE)

To find a simple high-breakdown regression estimator that shares the flexibility and nice asymptotic properties of the M-estimator, [20] introduced the S-estimate. The SE was developed as a robust alternative to traditional estimators like OLSE, particularly when dealing with data containing outliers or high-leverage points. They called it the S-estimate because it is derived from the M-scale estimate equation

$${\displaystyle \frac{1}{n}}\sum _{i=1}^n\rho \left({\displaystyle \frac{e_i\left(\mathit{\beta }\right)}{\widehat{\sigma }}}\right)=\delta .$$

(21)

In M-estimates, when $\sigma$ is unknown, we use Eq. (21) to obtain the scale parameter and regression estimates simultaneously. Let $\delta ={\mathbb{E}}_{\varphi }\left[\rho \left(x\right)\right]$, where $\varphi$ represents the standard normal density, and let

$$d\left(\mathbf{e}\right)={\displaystyle \frac{\#\{i:1\le i\le n,e_i=0\}}{n}}.$$

When $d\left(\mathbf{e}\right)<1-\delta /a$ (where a is the upper bound of $\rho$ and $a\in (0,\infty )$), the scale has a unique positive solution. If $d\left(\mathbf{e}\right)=1-\delta /a$, it may have infinite solutions including $\sigma =0$, and if $d\left(\mathbf{e}\right)>1-\delta /a$, then no solution exists.

To avoid such indeterminacies, we define that whenever $d\left(\mathbf{e}\right)\ge 1-\delta /a$, we set $\sigma \left(\mathbf{e}\right)=0$.

To define the S-estimator, we let $\rho$ satisfy the following condition

(A1)

$\rho$ is symmetric, continuously differentiable, and $\rho \left(0\right)=0$;

(A1)

There exists $c>0$ such that $\rho$ is strictly increasing on $[0,c]$ and constant on $[c,\infty ]$.

For each vector $\mathit{\beta }$, using Eq (21), we can calculate the dispersion of residuals $\widehat{\sigma }(e_1\left(\mathit{\beta }\right),\dots ,e_n\left(\mathit{\beta }\right))$, where $\rho$ satisfies (A1). Then the S-estimator $\widehat{\mathit{\beta }}$ is defined by

$$\widehat{\mathit{\beta }}=arg\underset{\mathit{\beta }}{min}\widehat{\sigma }(e_1\left(\mathit{\beta }\right),\dots ,e_n\left(\mathit{\beta }\right)).$$

(22)

2.1.7. The Modified Maximum Likelihood Estimator (MME)

The MM estimation is a special type of M-estimation developed by [21], and is particularly useful when dealing with non-normal data or outliers. MM estimation is a combination of high breakdown value estimation and efficient estimation. MM estimator was the first estimator with both a high breakdown point and high efficiency under normal error. The MM-estimator follows a three-stage procedure

Stage 1:

Compute an initial consistent robust estimate ${\widehat{\mathit{\beta }}}_0$ with high breakdown point (BP), possibly 50%, but not necessarily efficient.

Stage 2:

Compute the M-scale $\widehat{\sigma }$ of the residuals $e_i\left({\widehat{\mathit{\beta }}}_0\right)$ using Eq (21), with a function ${\rho }_0$ satisfying (A1) and choosing a constant $\delta$ such that $\delta /a=0.5$, where $a=sup{\rho }_0\left(e\right)$. Thus, the asymptotic BP of $\widehat{\sigma }$ is 0.5 [22].

Stage 3:

Let ${\rho }_1$ be another $\rho$-function satisfying (A1) such that

$$sup{\rho }_1\left(e\right)=sup{\rho }_0\left(e\right)=a,$$

(23)

and

$${\rho }_1\left(e\right)\le {\rho }_0\left(e\right).$$

(24)

Let ${\psi }_1={\rho }_1^{\prime }$ and define the objective function

$$L\left(\mathit{\beta }\right)=\sum _{i=1}^n{\rho }_1\left({\displaystyle \frac{e_i\left(\mathit{\beta }\right)}{\widehat{\sigma }}}\right),\mathrm{with}{\rho }_1(0/0)=0.$$

(25)

Then the MM-estimate ${\widehat{\mathit{\beta }}}_1$ is defined as any solution to

$$\sum _{i=1}^n{\psi }_1\left({\displaystyle \frac{e_i\left(\mathit{\beta }\right)}{\widehat{\sigma }}}\right)x_i=0,$$

(26)

that also satisfies

$$L\left({\widehat{\mathit{\beta }}}_1\right)\le L\left({\widehat{\mathit{\beta }}}_0\right).$$

(27)

Yohai [21] showed that any value of $\mathit{\beta }$ satisfying Eq. (26) and (27), for example a local minimum, will have the same efficiency as the global minimum, and its BP is no less than that of ${\widehat{\mathit{\beta }}}_0$. Thus, although the absolute minimum of $L\left(\mathit{\beta }\right)$ exists, it is not necessary to find it.

In the first stage, the robust initial estimate ${\widehat{\mathit{\beta }}}_0$ should satisfy regression, scale, and affine equivariance and have a high BP. LMS, LTS, and S-estimates are possible candidates.

For Stage 2, one way of choosing ${\rho }_0$ and ${\rho }_1$ is as follows: Let $\rho$ be a function satisfying (A1), and define

$${\rho }_0\left(e\right)=\rho \left({\displaystyle \frac{e}{k_0}}\right),{\rho }_1\left(e\right)=\rho \left({\displaystyle \frac{e}{k_1}}\right).$$

In order to satisfy Eq. (23), we must have $0<k_0<k_1$. The value of $k_0$ should be chosen such that $\delta /a=0.5$ holds.

2.2. Criteria for Assessing the Estimator’s Performance

Following a study by [23], two typical measures of accuracy for the estimator $\widehat{\theta }$ are the RMSE and the Bias. These measures can be estimated using bootstrap resampling. Specifically, given B bootstrap samples, the bootstrap estimate of Bias and RMSE is as follows

2.2.1. Bias Under the Bootstrap

$$\widehat{\mathrm{Bias}}={\displaystyle \frac{1}{B}}\sum _{b=1}^B{\widehat{\theta }}_b^{*}-\widehat{\theta }={\overline{\widehat{\theta }}}^{*}-\widehat{\theta }.$$

(28)

where ${\widehat{\theta }}_b^{*}$ are bootstrap copies of $\widehat{\theta }$.

2.2.2. RMSE Under the Bootstrap

$$\widehat{\mathrm{RMSE}}=\sqrt{{\displaystyle \frac{1}{B}}{\sum }_{b=1}^B{\left({\widehat{\theta }}_b^{*}-\widehat{\theta }\right)}^2}.$$

(29)

3. Monte Carlo Simulation Study

3.1. Simulation Procedure

The data generation process that follows was adapted from [24] and [25]. We generate n samples from the model using R studio

$$y_i={\beta }_0+{\beta }_1{X}_{i1}+{\beta }_2{X}_{i2}+{\beta }_3{X}_{i3}+{\epsilon }_i,i=1,2,\dots ,n$$

(30)

where the error terms are generated as ${\epsilon }_i\sim \mathcal{N}(0,1)$, and the explanatory variables are generated using the following equation

$$X_{ij}=Z_{ij}\sqrt{1-{\rho }^2}+\rho Z_{ij},i=1,2,\dots ,n,j=1,2,3,...,p$$

(31)

where $Z_{ij}$ are independent standard normal random numbers that are held fixed for a given sample of size n, and $\rho$ is the degree of multicollinearity between predictors. In this study, we consider three values of $\rho$ = 0.10, 0.50, and 0.98. The past study [1] mainly examined the RSE under high correlation; we extend the analysis to low and moderate levels to better understand its performance across varying degrees of multicollinearity.

The true values of the regression parameters are chosen as ${\beta }_0={\beta }_1=\dots ={\beta }_{p-1}=1$, which is a common restriction in simulation studies (see, for example, [1,13] and [26]). Additionally, the performance of the RSE is also evaluated for heavy-tailed error distributions, such as t-distributions and the Cauchy distribution.

Five different sample sizes are considered $n\in \{25,50,100,150,200\}$, with the number of predictors fixed at $p=3$, and different proportions of outliers are evaluated $\pi =0.00\%,10\%,25\%,50\%$. In each simulation setting, we perform $N=5,000$ MC replications, which is chosen as a compromise between achieving a low MC error and keeping the computation time reasonable.

To introduce outliers in the data, we consider two types, that is, y direction outliers and x direction outliers. For y direction outliers, we first generate the clean data according to equations (30) and (31) then randomly select $\lfloor n\times \pi \rfloor$ observations and replace their response values by adding a large constant, that is, $y_i^{\mathrm{outlier}}=y_i+10\times {\sigma }_{\epsilon }$, where ${\sigma }_{\epsilon }$ is the standard deviation of the error term. For x direction outliers, we randomly select $\lfloor n\times \pi \rfloor$ observations and replace their predictor values with $X_{ij}^{\mathrm{outlier}}=X_{ij}+10\times \mathrm{SD}\left(X_j\right)$, where $\mathrm{SD}\left(X_j\right)$ is the standard deviation of the j-th predictor.

A summary of the simulation scenarios is provided in Table 1. In addition, we evaluate the performance of the proposed estimators under the following ten cases

• Case I: ${\epsilon }_i\sim \mathcal{N}(0,1)$ - Standard normal distribution with 0.10, 0.50, and 0.98 multicollinearity and 0.00% outliers.
• Case II: ${\epsilon }_i\sim t_7$ — Student’s t-distribution with 7 degrees of freedom with 0.10, 0.50, and 0.98 multicollinearity and 0.00% outliers.
• Case III: ${\epsilon }_i\sim t_2$ — Student’s t-distribution with 2 degrees of freedom, 0.10, 0.50, and 0.98 multicollinearity and 0.00% outliers.
• Case IV: ${\epsilon }_i\sim \mathcal{C}(0,1)$ — Cauchy distribution errors with 0.10, 0.50, and 0.98 multicollinearity and 0.00% outliers.
• Case V: ${\epsilon }_i\sim \mathcal{N}(0,1)$ — Standard normal distribution with 10%, 25%, and 50% outliers with no multicollinearity.
• Case VI: ${\epsilon }_i\sim t_2$ — Student’s t-distribution with 2 degrees of freedom, 10%, 25%, and 50% outliers with no multicollinearity.
• Case VII: ${\epsilon }_i\sim \mathcal{N}(0,1)$ — 10%, 25%, and 50% outliers in the y-direction with 0.10, 0.50, and 0.98 multicollinearity.
• Case VIII: ${\epsilon }_i\sim \mathcal{N}(0,1)$ — 10%, 25%, and 50% outliers in the x-direction with 0.10, 0.50, and 0.98 multicollinearity.
• Case IX: ${\epsilon }_i\sim t_2$ (Student’s t-distribution with 2 degrees of freedom), with multicollinearity levels of 0.10, 0.50, and 0.98, and outliers in the y-direction at 10%, 25%, and 50%.
• Case X: ${\epsilon }_i\sim t_2$ (Student’s t-distribution with 2 degrees of freedom), with multicollinearity levels of 0.10, 0.50, and 0.98, and outliers in the x-direction at 10%, 25%, and 50%.

Table 1. Summary of Simulation Parameters Considered.

Sample size (n)	Multicollinearity ($\mathit{\rho }$)	Outlier ($\mathit{\pi }$)
25	0.10	0.00
50	0.50	0.10
100	0.98	0.25
150		0.50
200

Table 1. Summary of Simulation Parameters Considered.

Sample size (n)	Multicollinearity ($\mathit{\rho }$)	Outlier ($\mathit{\pi }$)
25	0.10	0.00
50	0.50	0.10
100	0.98	0.25
150		0.50
200

3.2. Simulation Study Findings

3.2.1. Performance Evaluation Findings Across the Cases

Case I (Table A3 and Figure 1): The JSE and RSE yield the smallest RMSE values across all sample sizes, consistent with their theoretical optimality properties. The MME, HME, OLSE, and LTSE also achieve RMSE values comparable to those of JSE and RSE only when $\rho$ = 0.50 and 0.98. In contrast, the LMSE and SE display relatively higher RMSEs, reflecting their lower efficiency in the presence of multicollinearity. The JSE and RSE have achieved exactly zero bias for low multicollinearity $\rho$ = 0.10 and 0.50 at sample sizes at $n\ge 100$, and LMSE shows the highest bias across nearly all scenarios. All estimators exhibit monotonic behavior, with RMSE values consistently decreasing as the sample size increases.

Case II (Table A4, and Figure 2): The RSE consistently achieves the lowest RMSE across all the conditions in this case. The JSE consistently achieves low bias across different sample sizes and multicollinearity levels. The LMSE has the overall worst performance across all the simulation scenarios.

Case III (Table A5 and Figure 3): In this case, the performance of the OLSE, JSE, and LMSE is substantially worse than that of any other robust estimators. The RSE, MME, and HME exhibit RMSE values that are almost similar, but the RSE is outperforming all the estimators in this case, as shown in Figure 3. The HME and MME consistently achieve low bias in almost all sample sizes and multicollinearity levels, and JSE has the highest bias at $\rho$ = 0.50 and 0.98.

Case IV(Table A6, and Figure 4): Under Cauchy-distributed errors, OLSE and JSE demonstrate erratic RMSE behavior, with values escalating due to the heavy-tailed nature of the distribution. In contrast, robust estimators maintain lower RMSEs and a monotonic decrease with increasing sample size, highlighting their stability. Furthermore, as the correlation coefficient ($\rho$) increases, the performance of OLSE and JSE worsens further. The RSE is the best estimator under the Cauchy distribution as shown by Figure 4. HME and MME continue to maintain a consistently lowest bias across all scenarios at $\rho$ = 0.98 and n = 200. OLSE and JSE are having the highest bias with values exceeding one.

Case V (Table A7, Table A8, and Figure 5, Figure 6): The MME performs the best under y direction, and LTSE performs the best under x direction. RSE yields a higher RMSE because it lacks robustness in both x and y directions. LTSE and SE maintain a small bias and low RMSE in the presence of x-outliers, whereas RSE, JSE, and OLSE break down due to high sensitivity, as shown in Figure 5 and Figure 6.

Case VI (Table A9, Table A10, and Figure 7, Figure 8):Table A9 and Figure 7 show that the MME, SE, HME, and LTSE consistently achieve the lowest RMSE and almost zero bias, making it the best performers, while the RSE, OLSE, and JSE perform the worst due to high RMSE values. Table A10 and Figure 8 indicate that the LTSE, MME, and SE are the most robust and accurate, while the RSE and JSE again show the poorest performance.

Case VII (Table A11, Table A12, Table A13, and Figure 9, Figure 10, Figure 11): When the data contains multicollinearity and outliers in the response direction, the performance of the OLSE, JSE, and LMSE worsens significantly, particularly as the percentage of outliers and the degree of multicollinearity increase as shown by Figure 9, Figure 10, and Figure 11. The following estimators, the LMSE, OLSE, and JSE in Figure 11, completely fail when 50% of the observations are contaminated. In contrast, the RSE, SE, HME, and MME demonstrate superior performance relative to OLSE, JSE, and LMSE under the same conditions. Moreover, the LTSE and SE exhibit even better robustness, maintaining substantially lower RMSE values compared to OLSE, JSE, and LMSE, particularly in the presence of low contamination, which is 10% in Figure 9. When there are 25% of outliers in Figure 10, we observe similar patterns with Figure 9. The RSE performs very well in cases of outliers and multicollinearity, followed by MME, HME, and LTSE.

Case VIII (Table A14, Table A15, Table A16, and Figure 12, Figure 13, Figure 14): The RSE is outperforming all the estimators in all scenarios when the data contains multicollinearity and the outliers in the x direction. The performance of LMSE, SE, LTSE, and OLSE worsens significantly as the percentage of outliers and multicollinearity increases, as shown by Figure 12, Figure 13, Figure 14. The bias is the smallest for SE and MME when $\pi =25\%$ and 50%.

Cases IX (Table A17, Table A18, Table A19, and Figure 15, Figure 16, Figure 17): The RSE consistently performs better than average performance among all the simulation scenarios. When 10% outliers and multicollinearity are present, the HME, MME, LTSE, and SE work with similar results as shown in Figure 15. The JSE and OLSE, in contrast, always give the poorest performance in all of these scenarios. LMSE shows less than ideal accuracy of performance as well, specifically under 10% contamination, over which it generates the largest bias values. Overall, the HME and MME are the most accurate estimators, with consistently lower bias compared to other methods.

Cases X (Table A20, Table A21, Table A22, and Figure 18, Figure 19, Figure 20): RMSE values are shown in Figure 18, Figure 19 and Figure 20 under different multicollinearity and x-outlier levels. The RMSE increases with both higher multicollinearity and larger proportions of x-outliers. RSE consistently provides the lowest RMSE, reflecting good stability, while OLSE and JSE perform the worst. In terms of bias, HME and MME both show the lowest bias in all scenarios, and JSE shows the highest bias. These findings also emphasize the better performance of RSE, and therefore its robustness and accuracy, and show the relative bias performance of the other estimators under challenging data conditions.

In summary, JSE and OLSE only work well when there are no outliers since it is very sensitive to outliers. The estimator’s bias and RMSE values were increased in the case of the multicollinearity degree ($\rho$) and outlier percentage ($\pi$), and decreased in the case of the sample size (n) being increased, when other factors were fixed. When only the multicollinearity problem existed in the model (as in Figure 1 and Figure 2, or when no outliers existed, i.e., $\pi =0.00\%$), the OLSE, JSE, HME, MME, and RSE were better than LMSE, LTSE, and SE. But in Figure 4, OLSE and JSE performed the worst, even though they had no outliers. When both problems existed (as in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, and Figure 14 or when outliers existed, i.e., $\pi >0.00\%$), the RSE, HME, and MME were better than the LMSE, LTSE, SE, JSE, and OLSE, respectively, for all $\pi$, $\rho$, and n values. Finally, the RSE achieved the best performance among all given estimators when outliers and multicollinearity exist.

4. Emprical Applications

In the previous section, we conducted an MC simulation study to compare the performance of the estimators. However, simulations are usually performed under some ideal conditions. In contrast to the MC simulation, this section considers three real linear regression datasets as an illustrative example in handling outliers and multicollinearity in linear regression. The datasets are the Milk dataset, the Real Estate Valuation dataset, and the Hawkins–Bradu–Kass dataset. We adopted a systematic trimming procedure in accordance with [27] to modify the outlier contamination levels in each dataset because real datasets frequently contain variable and uncontrollable levels of outliers. A direct comparison of theoretical, simulated, and real-data results is made possible by this method, which enables us to produce multiple versions of each dataset with particular outlier percentages that match our simulation study design. A summary of the datasets considered for this study is provided in Table A1.

Study 1: Milk Dataset

Milk dataset provided [28] is a composition of milk with 8 variables. The 8 variables are density, fat content, protein content, casein content, cheese dry substance measured in the factory, cheese dry substance measured in the laboratory, milk dry substance, and cheese produced. According to [28], the are 17 outliers in this dataset, which makes the percentage of outliers 20%. Observations (1^st–3^rd, 12^th, 13^th–17^th, 27^th, 41^st, 44^th, 47^th, 70^th, 74^th, 75^th and 77^th) are outliers.

4.1. Exploratory Analysis for Milk Dataset

4.1.1. Multicollinearity Detection for Milk Dataset

Table A2 presents VIF for milk dataset, which is a crucial diagnostic tool for detecting multicollinearity in regression analysis. The VIF quantifies how much the variance of a regression coefficient increases due to collinearity with other predictor variables. A VIF value greater than 10 indicates high multicollinearity. It is noted that from the milk dataset, variables $X_4$ and $X_5$ exhibit high VIF. Moreover, the observed condition number (CN) of 164.0314 indicates that strong multicollinearity exists among the regressors. It is observed from the Table 2 that some regressors ($X_2$,$X_3$,$X_4$,$X_5$) are highly correlated.

4.1.2. Outlier Detection Using Cook’s Distance for Milk Dataset

The Cook’s distance plot for the Milk dataset, with a threshold value of 0.047, is displayed in Figure 21. It was discovered that two observations (70 and 74) greatly surpass this threshold, with observation 70 achieving a value above 2.0. These data points are considered to be very significant and deserving of more investigation. Their existence indicates the possibility of anomalies or leverage points that might influence model predictions.

4.1.3. Testing for Normality for Milk Dataset

For this study, both theoretical and graphical techniques will be used to assess the normality of the OLS residuals. The Shapiro-Wilk (SW) test will be used as a theoretical tool. Figure 22 shows a diagnostic plot to assess residual normality in milk data. Visual inspection of these plots reveals that the Milk data displays approximately normal residuals. The results of the theoretical test yielded a p-value of 0.1964 ($p>0.05$), indicating that the milk data follows a normal distribution.

4.1.4. Model Fit and Evaluation for Milk Dataset

To compare the performance of the methods, the regression model was fitted using the Milk dataset for each method, considering the sparsity of the models. The bias and RMSE were finally used to evaluate how well the methods performed. The results presented in Table 3 cover the bias and the RMSE of each estimation method for the Milk dataset. The results show that the RSE has outperformed all the estimates when there are both outliers and multicollinearity. When there is multicollinearity and 0% outliers, the JSE, OLSE, and other robust estimates have almost the same RMSE values, the same thing that was happening in our simulation study in chapter four. The next estimators with similar performance when multicollinearity and outliers exist are JSE, MME, and HME. Figure 23 was used to assess the bias and RMSE. It was observed that the OLSE bias (red line) increases with increasing outlier percentage. The RSE RMSE values (black line) decrease with increasing outlier percentage.

Study 2 : Real Estate Valuation Dataset

Real estate valuation was collected in 2018 from Sindian District, New Taipei City, Taiwan, and it was recently published by [29]. The response variable in this study is the house price of unit area, measured in 10,000 New Taiwan Dollars per Ping, where one Ping corresponds to $3.3{\mathrm{m}}^2$. The explanatory variables include the transaction date ($X_1$), the house age in years ($X_2$), the distance to the nearest MRT station in meters ($X_3$), the number of convenience stores within walking distance ($X_4$), the geographic coordinate latitude in degrees ($X_5$), and the geographic coordinate longitude in degrees ($X_6$).

4.2. Exploratory Analysis for Real Estate Valuation

4.2.1. Multicollinearity Detection for Real Estate Valuation

The real estate valuation dataset exhibits low VIF in all the predictors; therefore, the CN of 14.7723 indicates that moderate multicollinearity exists among the predictors, as shown in Table A2. It is observed from the Table 4 that some regressors ($X_3$,$X_4$,$X_5$,$X_6$) are highly correlated.

4.2.2. Outlier Detection Using Cook’s Distance for Real Estate Valuation Dataset

Figure 24 displays the Cook’s distance plot with a 0.01 threshold for the real estate valuation dataset. In comparison to the other points, observation 271 shows a significantly higher Cook’s D value, indicating a significant impact on the regression results. Data points 149, 221, and 313 show additional moderate influences.

4.2.3. Testing for Normality for Real Estate Valuation Dataset

Figure 25 shows clear departures from normality due to skewed residual distributions. Therefore, the theoretical test confirms that the data is not normally distributed as the p-value is less than 0.05 ($p<0.0001$).

4.2.4. Model Fit and Evaluation for eal Estate Valuation Dataset

Table 5 presents the estimated bias and RMSE of various estimators under increasing levels of outlier contamination and multicollinearity. The RSE demonstrates the most consistent performance in terms of RMSE across all contamination levels, particularly excelling at higher outlier percentages. While JSE shows the lowest RMSE under clean data, its performance deteriorates significantly with more outliers. HME and OLSE maintain relatively low bias at all levels, but often at the cost of higher RMSE. Figure 26 was used to assess bias and RMSE. It was observed that HME, OLSE, and RSE maintain consistently low bias across all outlier levels. The RSE and JSE are the clear top performers with the low RMSE values. Despite the fact that RSE outperforms all other estimators in terms of RMSE, it exhibits higher bias compared to some of them. This indicates that while RSE achieves the best overall predictive accuracy by effectively balancing bias and variance, it does not perform best in terms of bias alone. Therefore, the RSE is the most outperforming estimator as outlier percentage increases, suggesting it has strong abilities to detect and manage multicollinearity and outliers.

4.3. Exploratory Analysis for Hawkins Bradu Kass Dataset

4.3.1. Multicollinearity Detection for Hawkins Bradu Kass Dataset

It is noteworthy that all predictors exhibit exceptionally high VIF values in Hawkins Bradu Kass dataset as shown in Table A2, and the observed CN of 102.5294 indicates that strong multicollinearity exists among the regressors. It is observed from the Table 6 that some regressors ($X_2$,$X_3$) are highly correlated.

4.3.2. Outlier Detection Using Cook’s Distance for Hawkins Bradu Kass Dataset

The Cook’s distance plot for the Hawkins Bradu Kass dataset, with a threshold value of 0.053, is displayed in Figure 27. Cook’s D values in observations 12 and 14 significantly surpass the threshold, highlighting the fact that they are as highly significant outliers. If not appropriately addressed, these points might bias the parameter estimates and overall model performance.

4.3.3. Testing for Normality for Hawkins Bradu Kass Dataset

Figure 28 shows clear departures from normality due to skewed residual distributions. Therefore, the theoretical test confirms that the data is not normally distributed as the p-value is less than 0.05 ($p<0.0001$).

4.3.4. Model Fit and Evaluation for Hawkins Bradu Kass Dataset.

As shown in Table 7 and Figure 29, in the absence of outliers, the JSE achieved the lowest RMSE, while HME had the lowest bias. Under 18.67% outlier contamination with multicollinearity, SE produced the lowest bias, and RSE achieved the lowest RMSE. MME also performed well under contamination, demonstrating both low bias and RMSE. In contrast, OLSE experienced a significant increase in both bias and RMSE when exposed to outliers. Overall, robust estimators like RSE and MME display greater stability and resilience under data contamination.

5. Discussion

The outcomes are consistent with the current literature regarding robust regression estimators. RSE’s superior performance in the presence of simultaneous multicollinearity and outlier contamination is attributed to its dual-component structure, that is, the HME component reduces the influence of outliers, while the JSE component maintains estimates through shrinkage, thereby alleviating variance inflation [1].

The MME performed better when outliers only appeared in the y-direction and there was no multicollinearity. This pattern is consistent with the findings of [30] and [31], who found that MME was the most effective estimator for pure vertical contamination. The failure of OLSE and JSE in the presence of Cauchy-distributed errors corroborates the conclusions of [32] and [33], which indicate that heavy-tailed distributions lead to extreme observations dominating the sum of squared residuals. One basic trade-off in robust estimation is demonstrated by the failure of high-breakdown estimators (LMSE, LTSE, SE) under multicollinearity without outliers [20]. This study’s overall findings support the idea that no single estimator is always the best. MME maintains advantages in environments dominated by y-direction outliers without multicollinearity [30,31,34], whereas RSE performs best when both multicollinearity and outliers are present [1]. These results highlight how crucial diagnostic analysis is prior to deciding on an estimation technique.

6. Concluding Remarks

This study evaluates the performance of RSE, which combines the shrinkage factors of ME and JSE, in the presence of multicollinearity and outliers in both the x and y directions. We evaluate the efficiency of RSE in an extensive MC simulation study with bias and RMSE criteria. As it stands, the simulation study is performed under several distributional conditions (normally, t-distributions, and Cauchy distributed errors), sample sizes, levels of multicollinearity, and outliers in both the x and y directions separately.

When multicollinearity and outliers in x and y directions exist, the RSE outperforms the classical OLSE, JSE, HME, MME, LMSE, LTSE, and SE. Additionally, when comparing the RSE to other existing robust estimators, we see that in the majority of the simulation scenarios evaluated, the RSE performs better. Based on the simulation study and its application to real datasets, we find that the RSE is the best estimator under conditions of multicollinearity and outliers in both the x and y directions. Future research may extend this work by comparing the RSE through simulation studies using various distributions, such as the Weibull, Gamma, and Poisson distributions, across different sample sizes, levels of outlier contamination, and degrees of multicollinearity.

Appendix A.1

Table A1. Summary of the Datasets.

Study No.	Dataset	n	p	No. of Outliers	Outliers (%)
	Milk dataset (original)	86	8	20	20%
1	Milk trimmed dataset	62	8	0	0%
	Milk trimmed dataset	81	8	8	10%
	Real Estate Valuation (original)	414	7	88	21.25%
2	Real Estate Valuation trimmed dataset	212	7	0	0%
	Real Estate Valuation trimmed dataset	268	7	41	15.3%
	Hawkins–Bradu–Kass dataset (original)	75	4	14	18.67%
3	Hawkins–Bradu–Kass trimmed dataset	61	4	0	0%

Table A1. Summary of the Datasets.

Study No.	Dataset	n	p	No. of Outliers	Outliers (%)
	Milk dataset (original)	86	8	20	20%
1	Milk trimmed dataset	62	8	0	0%
	Milk trimmed dataset	81	8	8	10%
	Real Estate Valuation (original)	414	7	88	21.25%
2	Real Estate Valuation trimmed dataset	212	7	0	0%
	Real Estate Valuation trimmed dataset	268	7	41	15.3%
	Hawkins–Bradu–Kass dataset (original)	75	4	14	18.67%
3	Hawkins–Bradu–Kass trimmed dataset	61	4	0	0%

Table A2. Variance Inflation Factor (VIF) of Milk Dataset, Real Estate Dataset, and Hawkins Bradu Kass Dataset.

Dataset	Variable	VIF	Variable	VIF
Milk dataset	$X_1$	2.2007	$X_5$	24.7561
	$X_2$	8.2865	$X_6$	3.2919
	$X_3$	7.2187	$X_8$	2.1834
	$X_4$	24.2253
Real Estate Valuation dataset	$X_1$	1.0147	$X_5$	1.6023
	$X_2$	1.0143	$X_6$	2.9263
	$X_3$	4.3230
	$X_4$	1.6170
Hawkins Bradu Kass datase	$X_1$	13.4320
	$X_2$	23.8535
	$X_3$	33.4325

Table A2. Variance Inflation Factor (VIF) of Milk Dataset, Real Estate Dataset, and Hawkins Bradu Kass Dataset.

Dataset	Variable	VIF	Variable	VIF
Milk dataset	$X_1$	2.2007	$X_5$	24.7561
	$X_2$	8.2865	$X_6$	3.2919
	$X_3$	7.2187	$X_8$	2.1834
	$X_4$	24.2253
Real Estate Valuation dataset	$X_1$	1.0147	$X_5$	1.6023
	$X_2$	1.0143	$X_6$	2.9263
	$X_3$	4.3230
	$X_4$	1.6170
Hawkins Bradu Kass datase	$X_1$	13.4320
	$X_2$	23.8535
	$X_3$	33.4325

Table A3. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Normal Distribution ${\epsilon }_i\sim N(0,1)$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0048	0.7969	0.0071	0.5625	0.0080	2.1962
	50	0.0031	0.5451	0.0064	0.3860	0.0173	1.5106
	100	0.0016	0.3808	0.0025	0.2692	0.0096	1.0585
	150	0.0019	0.3016	0.0010	0.2140	0.0036	0.8410
	200	0.0008	0.2651	0.0014	0.1833	0.0153	0.7204
HME	25	0.0004	0.7367	0.0055	0.5905	0.0193	2.3061
	50	0.0049	0.4955	0.0052	0.4067	0.0173	1.5944
	100	0.0021	0.3392	0.0009	0.2817	0.0050	1.1043
	150	0.0032	0.2705	0.0029	0.2245	0.0135	0.8834
	200	0.0027	0.2369	0.0016	0.1924	0.0152	0.7554
MME	25	0.0048	0.7450	0.0065	0.6064	0.0160	2.3551
	50	0.0034	0.4964	0.0064	0.4113	0.0191	1.6135
	100	0.0029	0.3450	0.0010	0.2885	0.0080	1.1326
	150	0.0032	0.2735	0.0029	0.2338	0.0181	0.9137
	200	0.0033	0.2377	0.0005	0.2031	0.0184	0.7736
LMSE	25	0.0236	1.5696	0.0125	1.4520	0.0490	5.6943
	50	0.0144	1.1159	0.0225	1.0589	0.0505	4.1858
	100	0.0114	0.8152	0.0045	0.8262	0.0335	3.1939
	150	0.0067	0.6859	0.0073	0.7339	0.0387	2.7792
	200	0.0087	0.6201	0.0094	0.6494	0.0063	2.4765
LTSE	25	0.0076	0.9591	0.0033	0.8883	0.0473	3.3716
	50	0.0038	0.5957	0.0080	0.5332	0.0150	2.0957
	100	0.0036	0.3777	0.0057	0.3466	0.0162	1.3443
	150	0.0043	0.3010	0.0013	0.2741	0.0124	1.0760
	200	0.0027	0.2567	0.0034	0.2336	0.0170	0.9105
SE	25	0.0135	1.1950	0.0122	1.1329	0.0784	4.3873
	50	0.0089	0.8378	0.0082	0.8368	0.0301	3.3277
	100	0.0099	0.5921	0.0028	0.6184	0.0254	2.3643
	150	0.0071	0.4814	0.0031	0.5198	0.0295	2.0266
	200	0.0027	0.4170	0.0037	0.4561	0.0173	1.7737
JSE	25	0.0037	0.2250	0.0107	0.4773	0.0661	1.9841
	50	0.0003	0.0370	0.0076	0.3276	0.0430	1.2924
	100	0.0000	0.0000	0.0046	0.2253	0.0434	0.8524
	150	0.0000	0.0000	0.0025	0.1809	0.0256	0.6505
	200	0.0000	0.0000	0.0024	0.1535	0.0194	0.5438
RSE	25	0.0037	0.1838	0.0116	0.5022	0.0759	2.0991
	50	0.0003	0.0235	0.0065	0.3444	0.0461	1.3762
	100	0.0000	0.0000	0.0055	0.2366	0.0425	0.8949
	150	0.0000	0.0000	0.0033	0.1896	0.0329	0.6874
	200	0.0000	0.0000	0.0028	0.1611	0.0233	0.5744

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A3. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Normal Distribution ${\epsilon }_i\sim N(0,1)$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0048	0.7969	0.0071	0.5625	0.0080	2.1962
	50	0.0031	0.5451	0.0064	0.3860	0.0173	1.5106
	100	0.0016	0.3808	0.0025	0.2692	0.0096	1.0585
	150	0.0019	0.3016	0.0010	0.2140	0.0036	0.8410
	200	0.0008	0.2651	0.0014	0.1833	0.0153	0.7204
HME	25	0.0004	0.7367	0.0055	0.5905	0.0193	2.3061
	50	0.0049	0.4955	0.0052	0.4067	0.0173	1.5944
	100	0.0021	0.3392	0.0009	0.2817	0.0050	1.1043
	150	0.0032	0.2705	0.0029	0.2245	0.0135	0.8834
	200	0.0027	0.2369	0.0016	0.1924	0.0152	0.7554
MME	25	0.0048	0.7450	0.0065	0.6064	0.0160	2.3551
	50	0.0034	0.4964	0.0064	0.4113	0.0191	1.6135
	100	0.0029	0.3450	0.0010	0.2885	0.0080	1.1326
	150	0.0032	0.2735	0.0029	0.2338	0.0181	0.9137
	200	0.0033	0.2377	0.0005	0.2031	0.0184	0.7736
LMSE	25	0.0236	1.5696	0.0125	1.4520	0.0490	5.6943
	50	0.0144	1.1159	0.0225	1.0589	0.0505	4.1858
	100	0.0114	0.8152	0.0045	0.8262	0.0335	3.1939
	150	0.0067	0.6859	0.0073	0.7339	0.0387	2.7792
	200	0.0087	0.6201	0.0094	0.6494	0.0063	2.4765
LTSE	25	0.0076	0.9591	0.0033	0.8883	0.0473	3.3716
	50	0.0038	0.5957	0.0080	0.5332	0.0150	2.0957
	100	0.0036	0.3777	0.0057	0.3466	0.0162	1.3443
	150	0.0043	0.3010	0.0013	0.2741	0.0124	1.0760
	200	0.0027	0.2567	0.0034	0.2336	0.0170	0.9105
SE	25	0.0135	1.1950	0.0122	1.1329	0.0784	4.3873
	50	0.0089	0.8378	0.0082	0.8368	0.0301	3.3277
	100	0.0099	0.5921	0.0028	0.6184	0.0254	2.3643
	150	0.0071	0.4814	0.0031	0.5198	0.0295	2.0266
	200	0.0027	0.4170	0.0037	0.4561	0.0173	1.7737
JSE	25	0.0037	0.2250	0.0107	0.4773	0.0661	1.9841
	50	0.0003	0.0370	0.0076	0.3276	0.0430	1.2924
	100	0.0000	0.0000	0.0046	0.2253	0.0434	0.8524
	150	0.0000	0.0000	0.0025	0.1809	0.0256	0.6505
	200	0.0000	0.0000	0.0024	0.1535	0.0194	0.5438
RSE	25	0.0037	0.1838	0.0116	0.5022	0.0759	2.0991
	50	0.0003	0.0235	0.0065	0.3444	0.0461	1.3762
	100	0.0000	0.0000	0.0055	0.2366	0.0425	0.8949
	150	0.0000	0.0000	0.0033	0.1896	0.0329	0.6874
	200	0.0000	0.0000	0.0028	0.1611	0.0233	0.5744

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A4. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Student’s t-distribution with 7 Degrees of Freedom ${\epsilon }_i\sim t_7$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0123	0.7426	0.0066	0.7806	0.0216	3.0798
	50	0.0027	0.4939	0.0037	0.5268	0.0152	2.0303
	100	0.0054	0.3541	0.0062	0.3753	0.0169	1.4894
	150	0.0010	0.2870	0.0053	0.3012	0.0153	1.1769
	200	0.0058	0.2611	0.0045	0.2817	0.0069	1.0910
HME	25	0.0061	0.6519	0.0154	0.6874	0.0306	2.7191
	50	0.0035	0.4402	0.0072	0.4624	0.0192	1.7788
	100	0.0039	0.3065	0.0040	0.3260	0.0102	1.2931
	150	0.0018	0.2491	0.0035	0.2613	0.0236	1.0218
	200	0.0019	0.2184	0.0017	0.2308	0.0067	0.9018
MME	25	0.0091	0.6554	0.0116	0.6922	0.0222	2.7395
	50	0.0020	0.4447	0.0037	0.4656	0.0166	1.7988
	100	0.0051	0.3109	0.0046	0.3317	0.0031	1.3056
	150	0.0043	0.2569	0.0053	0.2669	0.0206	1.0457
	200	0.0019	0.2233	0.0019	0.2338	0.0051	0.9262
LMSE	25	0.0018	1.4606	0.0085	1.5271	0.0909	6.0429
	50	0.0074	1.0703	0.0124	1.1522	0.0852	4.4395
	100	0.0114	0.8019	0.0070	0.8446	0.0392	3.3496
	150	0.0104	0.7081	0.0064	0.7417	0.0163	2.9055
	200	0.0085	0.6387	0.0037	0.6779	0.0252	2.6226
LTSE	25	0.0146	0.8923	0.0168	0.9548	0.0262	3.6888
	50	0.0030	0.5645	0.0082	0.6011	0.0218	2.3083
	100	0.0034	0.3680	0.0056	0.3962	0.0154	1.5443
	150	0.0027	0.3054	0.0027	0.3190	0.0260	1.2319
	200	0.0029	0.2662	0.0029	0.2745	0.0049	1.0743
SE	25	0.0149	1.0998	0.0063	1.1431	0.0280	4.6433
	50	0.0047	0.7914	0.0065	0.8444	0.0129	3.2186
	100	0.0058	0.5706	0.0094	0.6112	0.0277	2.3669
	150	0.0090	0.4875	0.0083	0.5134	0.0304	2.0260
	200	0.0046	0.4245	0.0056	0.4416	0.0107	1.7412
JSE	25	0.0234	0.6805	0.0246	0.6675	0.0777	2.8964
	50	0.0116	0.4579	0.0152	0.4494	0.0745	1.8283
	100	0.0063	0.3252	0.0068	0.3130	0.0524	1.2813
	150	0.0080	0.2689	0.0044	0.2548	0.0417	0.9715
	200	0.0027	0.2399	0.0010	0.2360	0.0345	0.9034
RSE	25	0.0157	0.5964	0.2093	0.5832	0.0781	2.5176
	50	0.0120	0.4152	0.0123	0.3979	0.0585	1.5641
	100	0.0049	0.2813	0.0055	0.2705	0.0484	1.0821
	150	0.0042	0.2349	0.0032	0.2212	0.0290	0.8186
	200	0.0014	0.2036	0.0008	0.1939	0.0286	0.7053

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A4. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Student’s t-distribution with 7 Degrees of Freedom ${\epsilon }_i\sim t_7$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0123	0.7426	0.0066	0.7806	0.0216	3.0798
	50	0.0027	0.4939	0.0037	0.5268	0.0152	2.0303
	100	0.0054	0.3541	0.0062	0.3753	0.0169	1.4894
	150	0.0010	0.2870	0.0053	0.3012	0.0153	1.1769
	200	0.0058	0.2611	0.0045	0.2817	0.0069	1.0910
HME	25	0.0061	0.6519	0.0154	0.6874	0.0306	2.7191
	50	0.0035	0.4402	0.0072	0.4624	0.0192	1.7788
	100	0.0039	0.3065	0.0040	0.3260	0.0102	1.2931
	150	0.0018	0.2491	0.0035	0.2613	0.0236	1.0218
	200	0.0019	0.2184	0.0017	0.2308	0.0067	0.9018
MME	25	0.0091	0.6554	0.0116	0.6922	0.0222	2.7395
	50	0.0020	0.4447	0.0037	0.4656	0.0166	1.7988
	100	0.0051	0.3109	0.0046	0.3317	0.0031	1.3056
	150	0.0043	0.2569	0.0053	0.2669	0.0206	1.0457
	200	0.0019	0.2233	0.0019	0.2338	0.0051	0.9262
LMSE	25	0.0018	1.4606	0.0085	1.5271	0.0909	6.0429
	50	0.0074	1.0703	0.0124	1.1522	0.0852	4.4395
	100	0.0114	0.8019	0.0070	0.8446	0.0392	3.3496
	150	0.0104	0.7081	0.0064	0.7417	0.0163	2.9055
	200	0.0085	0.6387	0.0037	0.6779	0.0252	2.6226
LTSE	25	0.0146	0.8923	0.0168	0.9548	0.0262	3.6888
	50	0.0030	0.5645	0.0082	0.6011	0.0218	2.3083
	100	0.0034	0.3680	0.0056	0.3962	0.0154	1.5443
	150	0.0027	0.3054	0.0027	0.3190	0.0260	1.2319
	200	0.0029	0.2662	0.0029	0.2745	0.0049	1.0743
SE	25	0.0149	1.0998	0.0063	1.1431	0.0280	4.6433
	50	0.0047	0.7914	0.0065	0.8444	0.0129	3.2186
	100	0.0058	0.5706	0.0094	0.6112	0.0277	2.3669
	150	0.0090	0.4875	0.0083	0.5134	0.0304	2.0260
	200	0.0046	0.4245	0.0056	0.4416	0.0107	1.7412
JSE	25	0.0234	0.6805	0.0246	0.6675	0.0777	2.8964
	50	0.0116	0.4579	0.0152	0.4494	0.0745	1.8283
	100	0.0063	0.3252	0.0068	0.3130	0.0524	1.2813
	150	0.0080	0.2689	0.0044	0.2548	0.0417	0.9715
	200	0.0027	0.2399	0.0010	0.2360	0.0345	0.9034
RSE	25	0.0157	0.5964	0.2093	0.5832	0.0781	2.5176
	50	0.0120	0.4152	0.0123	0.3979	0.0585	1.5641
	100	0.0049	0.2813	0.0055	0.2705	0.0484	1.0821
	150	0.0042	0.2349	0.0032	0.2212	0.0290	0.8186
	200	0.0014	0.2036	0.0008	0.1939	0.0286	0.7053

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A5. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Student’s t-distribution with 2 Degrees of Freedom ${\epsilon }_i\sim t_2$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0108	1.9081	0.0363	1.5837	0.0343	6.3193
	50	0.0083	0.9523	0.0063	0.9997	0.0226	3.9887
	100	0.0097	0.6049	0.0029	0.7770	0.0326	2.9553
	150	0.0068	0.5355	0.0029	0.5735	0.0283	2.2427
	200	0.0060	0.4347	0.0077	0.5177	0.0298	2.0001
HME	25	0.0069	0.7556	0.0040	0.7981	0.0368	3.1272
	50	0.0052	0.5005	0.0023	0.5157	0.0348	1.9727
	100	0.0036	0.3357	0.0036	0.3515	0.0179	1.3784
	150	0.0042	0.2731	0.0028	0.2860	0.0079	1.1151
	200	0.0024	0.2306	0.0031	0.2437	0.0014	0.9518
MME	25	0.0048	0.7392	0.0075	0.7847	0.0241	3.0678
	50	0.0067	0.4957	0.0030	0.5140	0.0348	1.9556
	100	0.0029	0.3305	0.0040	0.3492	0.0150	1.3666
	150	0.0026	0.2755	0.0019	0.2858	0.0109	1.1261
	200	0.0030	0.2283	0.0021	0.2438	0.0070	0.9481
LMSE	25	0.0371	1.4968	0.0141	1.5585	0.0470	6.1612
	50	0.0141	1.1147	0.0083	1.1513	0.0508	4.5106
	100	0.0085	0.8248	0.0105	0.8907	0.0191	3.4257
	150	0.0038	0.7199	0.0084	0.7672	0.0261	2.9467
	200	0.0042	0.6274	0.0075	0.6625	0.0196	2.5748
LTSE	25	0.0174	0.9393	0.0158	1.0102	0.0448	3.8620
	50	0.0033	0.6015	0.0092	0.6156	0.0173	2.3741
	100	0.0056	0.3893	0.0054	0.4122	0.0229	1.6104
	150	0.0054	0.3245	0.0042	0.3397	0.0138	1.3150
	200	0.0011	0.2687	0.0034	0.2872	0.0095	1.1192
SE	25	0.0152	1.1244	0.0097	1.2179	0.0888	4.6215
	50	0.0034	0.7851	0.0067	0.7939	0.0592	3.2794
	100	0.0048	0.5736	0.0075	0.6120	0.0133	2.3900
	150	0.0029	0.4841	0.0035	0.5061	0.0113	1.9832
	200	0.0018	0.3995	0.0048	0.4249	0.0141	1.6467
JSE	25	0.0457	1.3759	0.0470	1.6786	0.0724	6.2188
	50	0.0294	0.9000	0.0257	0.8909	0.0782	3.7598
	100	0.0118	0.5776	0.0199	0.7118	0.0897	2.8167
	150	0.0096	0.5067	0.0120	0.5562	0.0552	2.0879
	200	0.0056	0.4034	0.0130	0.4802	0.0643	1.8965
RSE	25	0.0239	0.6831	0.0210	0.6786	0.0681	2.9392
	50	0.0102	0.4820	0.0141	0.4488	0.0585	1.7551
	100	0.0040	0.3145	0.0105	0.2950	0.0537	1.1619
	150	0.0049	0.2565	0.0051	0.2422	0.0394	0.9057
	200	0.0018	0.2153	0.0018	0.2044	0.0307	0.7498

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A5. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Student’s t-distribution with 2 Degrees of Freedom ${\epsilon }_i\sim t_2$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0108	1.9081	0.0363	1.5837	0.0343	6.3193
	50	0.0083	0.9523	0.0063	0.9997	0.0226	3.9887
	100	0.0097	0.6049	0.0029	0.7770	0.0326	2.9553
	150	0.0068	0.5355	0.0029	0.5735	0.0283	2.2427
	200	0.0060	0.4347	0.0077	0.5177	0.0298	2.0001
HME	25	0.0069	0.7556	0.0040	0.7981	0.0368	3.1272
	50	0.0052	0.5005	0.0023	0.5157	0.0348	1.9727
	100	0.0036	0.3357	0.0036	0.3515	0.0179	1.3784
	150	0.0042	0.2731	0.0028	0.2860	0.0079	1.1151
	200	0.0024	0.2306	0.0031	0.2437	0.0014	0.9518
MME	25	0.0048	0.7392	0.0075	0.7847	0.0241	3.0678
	50	0.0067	0.4957	0.0030	0.5140	0.0348	1.9556
	100	0.0029	0.3305	0.0040	0.3492	0.0150	1.3666
	150	0.0026	0.2755	0.0019	0.2858	0.0109	1.1261
	200	0.0030	0.2283	0.0021	0.2438	0.0070	0.9481
LMSE	25	0.0371	1.4968	0.0141	1.5585	0.0470	6.1612
	50	0.0141	1.1147	0.0083	1.1513	0.0508	4.5106
	100	0.0085	0.8248	0.0105	0.8907	0.0191	3.4257
	150	0.0038	0.7199	0.0084	0.7672	0.0261	2.9467
	200	0.0042	0.6274	0.0075	0.6625	0.0196	2.5748
LTSE	25	0.0174	0.9393	0.0158	1.0102	0.0448	3.8620
	50	0.0033	0.6015	0.0092	0.6156	0.0173	2.3741
	100	0.0056	0.3893	0.0054	0.4122	0.0229	1.6104
	150	0.0054	0.3245	0.0042	0.3397	0.0138	1.3150
	200	0.0011	0.2687	0.0034	0.2872	0.0095	1.1192
SE	25	0.0152	1.1244	0.0097	1.2179	0.0888	4.6215
	50	0.0034	0.7851	0.0067	0.7939	0.0592	3.2794
	100	0.0048	0.5736	0.0075	0.6120	0.0133	2.3900
	150	0.0029	0.4841	0.0035	0.5061	0.0113	1.9832
	200	0.0018	0.3995	0.0048	0.4249	0.0141	1.6467
JSE	25	0.0457	1.3759	0.0470	1.6786	0.0724	6.2188
	50	0.0294	0.9000	0.0257	0.8909	0.0782	3.7598
	100	0.0118	0.5776	0.0199	0.7118	0.0897	2.8167
	150	0.0096	0.5067	0.0120	0.5562	0.0552	2.0879
	200	0.0056	0.4034	0.0130	0.4802	0.0643	1.8965
RSE	25	0.0239	0.6831	0.0210	0.6786	0.0681	2.9392
	50	0.0102	0.4820	0.0141	0.4488	0.0585	1.7551
	100	0.0040	0.3145	0.0105	0.2950	0.0537	1.1619
	150	0.0049	0.2565	0.0051	0.2422	0.0394	0.9057
	200	0.0018	0.2153	0.0018	0.2044	0.0307	0.7498

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A6. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Cauchy Distribution ${\epsilon }_i\sim C(0,1)$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	1.7105	410.0024	1.1075	458.8047	2.9021	1949.7800
	50	7.4097	88.0969	8.0337	95.2352	32.8347	366.0479
	100	1.1715	72.3916	1.2518	75.8250	5.6827	270.3350
	150	1.1942	72.8678	1.2158	65.0286	4.0633	269.6116
	200	1.0946	82.4840	1.3290	61.1405	5.1978	260.7603
HME	25	0.0202	1.0725	0.0131	1.1397	0.0682	4.4596
	50	0.0027	0.6690	0.0040	0.7065	0.0227	2.7626
	100	0.0043	0.4428	0.0035	0.4679	0.0198	1.8232
	150	0.0043	0.3481	0.0047	0.3659	0.0224	1.4325
	200	0.0026	0.2999	0.0017	0.3154	0.0026	1.2351
MME	25	0.0153	0.9846	0.0034	1.0397	0.0587	4.1478
	50	0.0027	0.6136	0.0051	0.6459	0.0446	2.5230
	100	0.0021	0.4140	0.0050	0.4379	0.0256	1.7081
	150	0.0020	0.3261	0.0035	0.3471	0.0118	1.3298
	200	0.0051	0.2766	0.0026	0.2908	0.0094	1.1309
LMSE	25	0.0167	1.6576	0.0167	1.7265	0.0770	6.7443
	50	0.0102	1.1152	0.0102	1.1698	0.0990	4.5277
	100	0.0073	0.7885	0.0113	0.8272	0.0224	3.3110
	150	0.0041	0.6602	0.0030	0.6842	0.0299	2.6303
	200	0.0044	0.5764	0.0022	0.6247	0.0366	2.4631
LTSE	25	0.0117	1.1359	0.0084	1.1692	0.0235	4.5503
	50	0.0109	0.6670	0.0056	0.6998	0.0437	2.7133
	100	0.0032	0.4465	0.0061	0.4661	0.0118	1.8315
	150	0.0046	0.3569	0.0047	0.3711	0.0232	1.4465
	200	0.0046	0.3029	0.0041	0.3189	0.0127	1.2497
SE	25	0.0097	1.2065	0.0132	1.2566	0.0738	4.9714
	50	0.0149	0.7895	0.0098	0.8189	0.0270	3.1879
	100	0.0067	0.5307	0.0087	0.5537	0.0319	2.1612
	150	0.0102	0.4192	0.0070	0.4375	0.0333	1.7050
	200	0.0039	0.3603	0.0058	0.3829	0.0182	1.4984
JSE	25	1.5931	409.9958	0.9673	458.8019	2.9194	1949.7800
	50	7.3356	88.0764	7.9925	95.2344	32.7987	270.3315
	100	1.1244	82.4731	1.2966	75.8101	5.7051	366.0451
	150	1.1460	72.8492	1.1730	65.0195	4.0186	269.6087
	200	1.0538	72.3765	1.3648	61.1171	5.2656	260.7565
RSE	25	0.0321	0.9482	0.0383	0.9908	0.0662	4.3078
	50	0.0182	0.6170	0.0232	0.6077	0.0801	2.5756
	100	0.0037	0.4146	0.0126	0.4003	0.0822	1.6125
	150	0.0024	0.3266	0.0047	0.3100	0.0594	1.2199
	200	0.0049	0.2803	0.0092	0.2658	0.0444	1.0249

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A6. Estimated Bias and RMSE Values with Multicollinearity and no Outliers for Cauchy Distribution ${\epsilon }_i\sim C(0,1)$.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	1.7105	410.0024	1.1075	458.8047	2.9021	1949.7800
	50	7.4097	88.0969	8.0337	95.2352	32.8347	366.0479
	100	1.1715	72.3916	1.2518	75.8250	5.6827	270.3350
	150	1.1942	72.8678	1.2158	65.0286	4.0633	269.6116
	200	1.0946	82.4840	1.3290	61.1405	5.1978	260.7603
HME	25	0.0202	1.0725	0.0131	1.1397	0.0682	4.4596
	50	0.0027	0.6690	0.0040	0.7065	0.0227	2.7626
	100	0.0043	0.4428	0.0035	0.4679	0.0198	1.8232
	150	0.0043	0.3481	0.0047	0.3659	0.0224	1.4325
	200	0.0026	0.2999	0.0017	0.3154	0.0026	1.2351
MME	25	0.0153	0.9846	0.0034	1.0397	0.0587	4.1478
	50	0.0027	0.6136	0.0051	0.6459	0.0446	2.5230
	100	0.0021	0.4140	0.0050	0.4379	0.0256	1.7081
	150	0.0020	0.3261	0.0035	0.3471	0.0118	1.3298
	200	0.0051	0.2766	0.0026	0.2908	0.0094	1.1309
LMSE	25	0.0167	1.6576	0.0167	1.7265	0.0770	6.7443
	50	0.0102	1.1152	0.0102	1.1698	0.0990	4.5277
	100	0.0073	0.7885	0.0113	0.8272	0.0224	3.3110
	150	0.0041	0.6602	0.0030	0.6842	0.0299	2.6303
	200	0.0044	0.5764	0.0022	0.6247	0.0366	2.4631
LTSE	25	0.0117	1.1359	0.0084	1.1692	0.0235	4.5503
	50	0.0109	0.6670	0.0056	0.6998	0.0437	2.7133
	100	0.0032	0.4465	0.0061	0.4661	0.0118	1.8315
	150	0.0046	0.3569	0.0047	0.3711	0.0232	1.4465
	200	0.0046	0.3029	0.0041	0.3189	0.0127	1.2497
SE	25	0.0097	1.2065	0.0132	1.2566	0.0738	4.9714
	50	0.0149	0.7895	0.0098	0.8189	0.0270	3.1879
	100	0.0067	0.5307	0.0087	0.5537	0.0319	2.1612
	150	0.0102	0.4192	0.0070	0.4375	0.0333	1.7050
	200	0.0039	0.3603	0.0058	0.3829	0.0182	1.4984
JSE	25	1.5931	409.9958	0.9673	458.8019	2.9194	1949.7800
	50	7.3356	88.0764	7.9925	95.2344	32.7987	270.3315
	100	1.1244	82.4731	1.2966	75.8101	5.7051	366.0451
	150	1.1460	72.8492	1.1730	65.0195	4.0186	269.6087
	200	1.0538	72.3765	1.3648	61.1171	5.2656	260.7565
RSE	25	0.0321	0.9482	0.0383	0.9908	0.0662	4.3078
	50	0.0182	0.6170	0.0232	0.6077	0.0801	2.5756
	100	0.0037	0.4146	0.0126	0.4003	0.0822	1.6125
	150	0.0024	0.3266	0.0047	0.3100	0.0594	1.2199
	200	0.0049	0.2803	0.0092	0.2658	0.0444	1.0249

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A7. Estimated Bias and RMSE Values for Normal Errors with Outliers in the y Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0055	0.6950	0.0071	0.9077	0.0014	0.9806
	50	0.0012	0.6690	0.0026	0.6134	0.0041	0.9072
	100	0.0019	0.6467	0.0003	0.4100	0.0031	0.8761
	150	0.0020	0.6210	0.0035	0.3268	0.0042	0.8328
	200	0.0010	0.6007	0.0020	0.2863	0.0042	0.7854
HME	25	0.0031	0.2733	0.0035	0.3405	0.0041	0.3394
	50	0.0009	0.1834	0.0007	0.2106	0.0002	0.2116
	100	0.0021	0.1246	0.0004	0.1218	0.0012	0.1384
	150	0.0005	0.0999	0.0001	0.1091	0.0002	0.1100
	200	0.0004	0.0863	0.0010	0.0948	0.0003	0.0876
MME	25	0.0024	0.2521	0.0031	0.2543	0.0014	0.2838
	50	0.0015	0.1675	0.0000	0.2099	0.0019	0.1807
	100	0.0004	0.1143	0.0002	0.1180	0.0018	0.1213
	150	0.0003	0.0917	0.0004	0.1001	0.0004	0.0974
	200	0.0007	0.0795	0.0080	0.0846	0.0003	0.0734
LMSE	25	0.0024	0.5255	0.0004	0.5347	0.0013	0.5462
	50	0.0004	0.3570	0.0012	0.3591	0.0078	0.3599
	100	0.0007	0.2549	0.0005	0.2547	0.0026	0.2594
	150	0.0016	0.2171	0.0007	0.2150	0.0013	0.2177
	200	0.0028	0.1946	0.0017	0.1924	0.0009	0.1914
LTSE	25	0.0020	0.3909	0.0021	0.3222	0.0021	0.3264
	50	0.0010	0.1966	0.0007	0.2367	0.0029	0.2000
	100	0.0005	0.1268	0.0007	0.1674	0.0018	0.1309
	150	0.0004	0.1008	0.0003	0.1388	0.0002	0.1033
	200	0.0007	0.0865	0.0004	0.1094	0.0005	0.0906
SE	25	0.0020	0.3909	0.0008	0.3831	0.0020	0.3865
	50	0.0006	0.2670	0.0015	0.2588	0.0049	0.2597
	100	0.0005	0.1817	0.0017	0.1786	0.0007	0.1805
	150	0.0004	0.1477	0.0009	0.1438	0.0006	0.1457
	200	0.0013	0.1277	0.0003	0.1239	0.0007	0.1224
JSE	25	0.2602	0.6801	0.3516	0.7350	0.3477	0.7329
	50	0.2062	0.6573	0.0263	0.5623	0.2685	0.7280
	100	0.0123	0.6312	0.0168	0.4063	0.1634	0.7044
	150	0.0004	0.6011	0.1233	0.3295	0.1242	0.6825
	200	0.0672	0.5698	0.0981	0.2892	0.0957	0.6657
RSE	25	0.4704	0.6585	0.4667	0.8030	0.4757	0.6995
	50	0.4905	0.6355	0.0487	0.5328	0.4881	0.6709
	100	0.0495	0.5889	0.0493	0.3510	0.4898	0.6538
	150	0.0493	0.5504	0.4951	0.2790	0.4943	0.6222
	200	0.4947	0.5065	0.4935	0.2581	0.4974	0.6098

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A7. Estimated Bias and RMSE Values for Normal Errors with Outliers in the y Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0055	0.6950	0.0071	0.9077	0.0014	0.9806
	50	0.0012	0.6690	0.0026	0.6134	0.0041	0.9072
	100	0.0019	0.6467	0.0003	0.4100	0.0031	0.8761
	150	0.0020	0.6210	0.0035	0.3268	0.0042	0.8328
	200	0.0010	0.6007	0.0020	0.2863	0.0042	0.7854
HME	25	0.0031	0.2733	0.0035	0.3405	0.0041	0.3394
	50	0.0009	0.1834	0.0007	0.2106	0.0002	0.2116
	100	0.0021	0.1246	0.0004	0.1218	0.0012	0.1384
	150	0.0005	0.0999	0.0001	0.1091	0.0002	0.1100
	200	0.0004	0.0863	0.0010	0.0948	0.0003	0.0876
MME	25	0.0024	0.2521	0.0031	0.2543	0.0014	0.2838
	50	0.0015	0.1675	0.0000	0.2099	0.0019	0.1807
	100	0.0004	0.1143	0.0002	0.1180	0.0018	0.1213
	150	0.0003	0.0917	0.0004	0.1001	0.0004	0.0974
	200	0.0007	0.0795	0.0080	0.0846	0.0003	0.0734
LMSE	25	0.0024	0.5255	0.0004	0.5347	0.0013	0.5462
	50	0.0004	0.3570	0.0012	0.3591	0.0078	0.3599
	100	0.0007	0.2549	0.0005	0.2547	0.0026	0.2594
	150	0.0016	0.2171	0.0007	0.2150	0.0013	0.2177
	200	0.0028	0.1946	0.0017	0.1924	0.0009	0.1914
LTSE	25	0.0020	0.3909	0.0021	0.3222	0.0021	0.3264
	50	0.0010	0.1966	0.0007	0.2367	0.0029	0.2000
	100	0.0005	0.1268	0.0007	0.1674	0.0018	0.1309
	150	0.0004	0.1008	0.0003	0.1388	0.0002	0.1033
	200	0.0007	0.0865	0.0004	0.1094	0.0005	0.0906
SE	25	0.0020	0.3909	0.0008	0.3831	0.0020	0.3865
	50	0.0006	0.2670	0.0015	0.2588	0.0049	0.2597
	100	0.0005	0.1817	0.0017	0.1786	0.0007	0.1805
	150	0.0004	0.1477	0.0009	0.1438	0.0006	0.1457
	200	0.0013	0.1277	0.0003	0.1239	0.0007	0.1224
JSE	25	0.2602	0.6801	0.3516	0.7350	0.3477	0.7329
	50	0.2062	0.6573	0.0263	0.5623	0.2685	0.7280
	100	0.0123	0.6312	0.0168	0.4063	0.1634	0.7044
	150	0.0004	0.6011	0.1233	0.3295	0.1242	0.6825
	200	0.0672	0.5698	0.0981	0.2892	0.0957	0.6657
RSE	25	0.4704	0.6585	0.4667	0.8030	0.4757	0.6995
	50	0.4905	0.6355	0.0487	0.5328	0.4881	0.6709
	100	0.0495	0.5889	0.0493	0.3510	0.4898	0.6538
	150	0.0493	0.5504	0.4951	0.2790	0.4943	0.6222
	200	0.4947	0.5065	0.4935	0.2581	0.4974	0.6098

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A8. Estimated Bias and RMSE Values under Normal Errors with Outliers in the x Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.1225	0.6138	0.3650	0.6569	0.5100	0.7285
	50	0.2247	0.5014	0.3927	0.5589	0.5224	0.6899
	100	0.2599	0.4067	0.4165	0.4371	0.5259	0.6514
	150	0.2600	0.3993	0.4293	0.4270	0.5167	0.6235
	200	0.2769	0.3824	0.4227	0.4136	0.5269	0.6233
HME	25	0.1103	0.4643	0.3445	0.4789	0.5015	0.6688
	50	0.1681	0.4031	0.3641	0.4104	0.5229	0.6262
	100	0.1824	0.2849	0.3908	0.3541	0.5237	0.5897
	150	0.1872	0.2604	0.4029	0.3105	0.5169	0.5801
	200	0.1901	0.2482	0.3968	0.3025	0.5242	0.5755
MME	25	0.0285	0.4011	0.1932	0.4098	0.4868	0.6631
	50	0.0513	0.3470	0.2401	0.3560	0.5219	0.6192
	100	0.0567	0.2262	0.3271	0.2682	0.5213	0.5883
	150	0.0619	0.2060	0.3510	0.2083	0.5162	0.5775
	200	0.0646	0.1801	0.3523	0.1789	0.5242	0.5730
LMSE	25	0.0078	0.5375	0.1014	0.5375	0.2804	0.8372
	50	0.0173	0.3772	0.0688	0.3772	0.3846	0.6487
	100	0.0106	0.2467	0.0601	0.2867	0.3937	0.5898
	150	0.0127	0.2160	0.0571	0.2142	0.4303	0.5810
	200	0.0116	0.1877	0.0428	0.1930	0.4311	0.5770
LTSE	25	0.0154	0.3907	0.1135	0.3907	0.3341	0.6326
	50	0.0263	0.2727	0.0953	0.2727	0.4087	0.5581
	100	0.0245	0.1806	0.0964	0.2048	0.4324	0.5398
	150	0.0251	0.1508	0.1036	0.1539	0.4324	0.5242
	200	0.0271	0.1358	0.0884	0.1487	0.4424	0.5140
SE	25	0.0129	0.4191	0.1048	0.4191	0.3425	0.7257
	50	0.0170	0.2859	0.0757	0.2859	0.4262	0.6250
	100	0.0139	0.1775	0.0613	0.2056	0.4381	0.5781
	150	0.0128	0.1548	0.0591	0.1542	0.4823	0.5607
	200	0.0135	0.1294	0.0464	0.1453	0.4764	0.5479
JSE	25	0.1912	0.6128	0.4938	0.6128	0.7283	0.8096
	50	0.2693	0.5501	0.4705	0.5501	0.6711	0.7674
	100	0.2874	0.4417	0.4604	0.5021	0.6127	0.7350
	150	0.2874	0.4274	0.4605	0.4684	0.5777	0.7033
	200	0.2894	0.4020	0.4458	0.4554	0.5764	0.6893
RSE	25	0.1303	0.5878	0.4913	0.5870	0.8224	0.7151
	50	0.0528	0.4982	0.5671	0.4980	0.8589	0.6823
	100	0.1036	0.4054	0.6383	0.4365	0.8633	0.6389
	150	0.1213	0.3847	0.6688	0.4149	0.8607	0.6198
	200	0.1324	0.3772	0.6593	0.4039	0.8738	0.6198

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A8. Estimated Bias and RMSE Values under Normal Errors with Outliers in the x Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.1225	0.6138	0.3650	0.6569	0.5100	0.7285
	50	0.2247	0.5014	0.3927	0.5589	0.5224	0.6899
	100	0.2599	0.4067	0.4165	0.4371	0.5259	0.6514
	150	0.2600	0.3993	0.4293	0.4270	0.5167	0.6235
	200	0.2769	0.3824	0.4227	0.4136	0.5269	0.6233
HME	25	0.1103	0.4643	0.3445	0.4789	0.5015	0.6688
	50	0.1681	0.4031	0.3641	0.4104	0.5229	0.6262
	100	0.1824	0.2849	0.3908	0.3541	0.5237	0.5897
	150	0.1872	0.2604	0.4029	0.3105	0.5169	0.5801
	200	0.1901	0.2482	0.3968	0.3025	0.5242	0.5755
MME	25	0.0285	0.4011	0.1932	0.4098	0.4868	0.6631
	50	0.0513	0.3470	0.2401	0.3560	0.5219	0.6192
	100	0.0567	0.2262	0.3271	0.2682	0.5213	0.5883
	150	0.0619	0.2060	0.3510	0.2083	0.5162	0.5775
	200	0.0646	0.1801	0.3523	0.1789	0.5242	0.5730
LMSE	25	0.0078	0.5375	0.1014	0.5375	0.2804	0.8372
	50	0.0173	0.3772	0.0688	0.3772	0.3846	0.6487
	100	0.0106	0.2467	0.0601	0.2867	0.3937	0.5898
	150	0.0127	0.2160	0.0571	0.2142	0.4303	0.5810
	200	0.0116	0.1877	0.0428	0.1930	0.4311	0.5770
LTSE	25	0.0154	0.3907	0.1135	0.3907	0.3341	0.6326
	50	0.0263	0.2727	0.0953	0.2727	0.4087	0.5581
	100	0.0245	0.1806	0.0964	0.2048	0.4324	0.5398
	150	0.0251	0.1508	0.1036	0.1539	0.4324	0.5242
	200	0.0271	0.1358	0.0884	0.1487	0.4424	0.5140
SE	25	0.0129	0.4191	0.1048	0.4191	0.3425	0.7257
	50	0.0170	0.2859	0.0757	0.2859	0.4262	0.6250
	100	0.0139	0.1775	0.0613	0.2056	0.4381	0.5781
	150	0.0128	0.1548	0.0591	0.1542	0.4823	0.5607
	200	0.0135	0.1294	0.0464	0.1453	0.4764	0.5479
JSE	25	0.1912	0.6128	0.4938	0.6128	0.7283	0.8096
	50	0.2693	0.5501	0.4705	0.5501	0.6711	0.7674
	100	0.2874	0.4417	0.4604	0.5021	0.6127	0.7350
	150	0.2874	0.4274	0.4605	0.4684	0.5777	0.7033
	200	0.2894	0.4020	0.4458	0.4554	0.5764	0.6893
RSE	25	0.1303	0.5878	0.4913	0.5870	0.8224	0.7151
	50	0.0528	0.4982	0.5671	0.4980	0.8589	0.6823
	100	0.1036	0.4054	0.6383	0.4365	0.8633	0.6389
	150	0.1213	0.3847	0.6688	0.4149	0.8607	0.6198
	200	0.1324	0.3772	0.6593	0.4039	0.8738	0.6198

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A9. Estimated Bias and RMSE under ${\epsilon }_i\sim t_2$ with Outliers in the y Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0457	0.8724	0.0108	0.9139	0.3930	1.0740
	50	0.0267	0.5982	0.0230	0.6246	0.3705	0.8713
	100	0.0233	0.4566	0.0136	0.5075	0.4457	0.7181
	150	0.0188	0.3505	0.0276	0.3837	0.3872	0.6936
	200	0.0189	0.3341	0.0043	0.3400	0.4038	0.6850
ME	25	0.0208	0.4170	0.0102	0.4597	0.1766	0.8102
	50	0.0031	0.2449	0.0197	0.2570	0.1835	0.5907
	100	0.0141	0.1654	0.0059	0.1681	0.1964	0.4931
	150	0.0029	0.1322	0.0024	0.1516	0.2044	0.4554
	200	0.0021	0.1266	0.0037	0.1169	0.2053	0.4281
MME	25	0.0022	0.3946	0.0113	0.4170	0.0299	0.6781
	50	0.0008	0.2122	0.0235	0.2452	0.0475	0.4995
	100	0.0123	0.1573	0.0009	0.1612	0.0993	0.4100
	150	0.0044	0.1205	0.0007	0.1362	0.0906	0.3616
	200	0.0039	0.1158	0.0042	0.1057	0.0901	0.3397
LMSE	25	0.0123	0.5972	0.0260	0.6250	0.0345	0.6433
	50	0.0217	0.4033	0.0009	0.3650	0.0146	0.4197
	100	0.0028	0.2630	0.0049	0.2685	0.0317	0.3096
	150	0.0075	0.2153	0.0102	0.2092	0.0140	0.2627
	200	0.0053	0.1956	0.0122	0.1731	0.0033	0.2433
LTSE	25	0.0081	0.4295	0.0103	0.4563	0.0157	0.4986
	50	0.0032	0.2272	0.0203	0.2466	0.0226	0.3812
	100	0.0097	0.1658	0.0007	0.1595	0.0435	0.2623
	150	0.0069	0.1249	0.0036	0.1300	0.0199	0.2243
	200	0.0076	0.1155	0.0022	0.1046	0.0311	0.1865
SE	25	0.0130	0.4157	0.0041	0.4847	0.0254	0.5413
	50	0.0024	0.2574	0.0230	0.2676	0.0375	0.3912
	100	0.0061	0.1817	0.0040	0.1702	0.0580	0.2859
	150	0.0126	0.1366	0.0042	0.1362	0.0317	0.2358
	200	0.0055	0.1233	0.0027	0.1084	0.0493	0.2046
JSE	25	0.2578	0.6665	0.3083	0.7295	0.6597	0.9887
	50	0.2344	0.5465	0.2726	0.5550	0.7327	0.9186
	100	0.2220	0.4315	0.2031	0.4319	0.7492	0.8692
	150	0.1064	0.3455	0.0935	0.3493	0.6862	0.8416
	200	0.0805	0.3229	0.1119	0.3013	0.6758	0.8293
RSE	25	0.0204	0.7605	0.0753	0.8116	0.2687	1.1759
	50	0.0109	0.5665	0.0525	0.5923	0.4567	0.8200
	100	0.0487	0.4300	0.0482	0.4537	0.5525	0.6905
	150	0.0068	0.3458	0.0151	0.3743	0.6033	0.6707
	200	0.0091	0.3282	0.0136	0.3082	0.6156	0.6582

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A9. Estimated Bias and RMSE under ${\epsilon }_i\sim t_2$ with Outliers in the y Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0457	0.8724	0.0108	0.9139	0.3930	1.0740
	50	0.0267	0.5982	0.0230	0.6246	0.3705	0.8713
	100	0.0233	0.4566	0.0136	0.5075	0.4457	0.7181
	150	0.0188	0.3505	0.0276	0.3837	0.3872	0.6936
	200	0.0189	0.3341	0.0043	0.3400	0.4038	0.6850
ME	25	0.0208	0.4170	0.0102	0.4597	0.1766	0.8102
	50	0.0031	0.2449	0.0197	0.2570	0.1835	0.5907
	100	0.0141	0.1654	0.0059	0.1681	0.1964	0.4931
	150	0.0029	0.1322	0.0024	0.1516	0.2044	0.4554
	200	0.0021	0.1266	0.0037	0.1169	0.2053	0.4281
MME	25	0.0022	0.3946	0.0113	0.4170	0.0299	0.6781
	50	0.0008	0.2122	0.0235	0.2452	0.0475	0.4995
	100	0.0123	0.1573	0.0009	0.1612	0.0993	0.4100
	150	0.0044	0.1205	0.0007	0.1362	0.0906	0.3616
	200	0.0039	0.1158	0.0042	0.1057	0.0901	0.3397
LMSE	25	0.0123	0.5972	0.0260	0.6250	0.0345	0.6433
	50	0.0217	0.4033	0.0009	0.3650	0.0146	0.4197
	100	0.0028	0.2630	0.0049	0.2685	0.0317	0.3096
	150	0.0075	0.2153	0.0102	0.2092	0.0140	0.2627
	200	0.0053	0.1956	0.0122	0.1731	0.0033	0.2433
LTSE	25	0.0081	0.4295	0.0103	0.4563	0.0157	0.4986
	50	0.0032	0.2272	0.0203	0.2466	0.0226	0.3812
	100	0.0097	0.1658	0.0007	0.1595	0.0435	0.2623
	150	0.0069	0.1249	0.0036	0.1300	0.0199	0.2243
	200	0.0076	0.1155	0.0022	0.1046	0.0311	0.1865
SE	25	0.0130	0.4157	0.0041	0.4847	0.0254	0.5413
	50	0.0024	0.2574	0.0230	0.2676	0.0375	0.3912
	100	0.0061	0.1817	0.0040	0.1702	0.0580	0.2859
	150	0.0126	0.1366	0.0042	0.1362	0.0317	0.2358
	200	0.0055	0.1233	0.0027	0.1084	0.0493	0.2046
JSE	25	0.2578	0.6665	0.3083	0.7295	0.6597	0.9887
	50	0.2344	0.5465	0.2726	0.5550	0.7327	0.9186
	100	0.2220	0.4315	0.2031	0.4319	0.7492	0.8692
	150	0.1064	0.3455	0.0935	0.3493	0.6862	0.8416
	200	0.0805	0.3229	0.1119	0.3013	0.6758	0.8293
RSE	25	0.0204	0.7605	0.0753	0.8116	0.2687	1.1759
	50	0.0109	0.5665	0.0525	0.5923	0.4567	0.8200
	100	0.0487	0.4300	0.0482	0.4537	0.5525	0.6905
	150	0.0068	0.3458	0.0151	0.3743	0.6033	0.6707
	200	0.0091	0.3282	0.0136	0.3082	0.6156	0.6582

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A10. Estimated Bias and RMSE for ${\epsilon }_i\sim t_2$ with Outliers in the x Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		Bias	RMSE	Bias	RMSE	Bias	RMSE
OLSE	25	0.2460	0.6138	0.2460	0.8977	0.5062	0.7285
	50	0.3284	0.5014	0.3284	0.6730	0.5376	0.6899
	100	0.2551	0.4067	0.3381	0.5003	0.5133	0.6504
	150	0.2546	0.4000	0.3364	0.4804	0.5158	0.6086
	200	0.2783	0.3824	0.3544	0.4236	0.5260	0.5897
ME	25	0.2329	0.4643	0.2329	0.4745	0.4999	0.6688
	50	0.2970	0.4031	0.2970	0.4299	0.5360	0.6262
	100	0.2078	0.2849	0.2898	0.3909	0.5190	0.5897
	150	0.2336	0.2798	0.2886	0.3667	0.5216	0.5601
	200	0.2234	0.2682	0.3060	0.3500	0.5271	0.5455
MME	25	0.1196	0.4011	0.1196	0.4299	0.4886	0.6631
	50	0.1961	0.3470	0.1961	0.3902	0.5335	0.6092
	100	0.1135	0.2262	0.1969	0.3339	0.5159	0.5383
	150	0.1231	0.2060	0.2112	0.2945	0.5205	0.5175
	200	0.1286	0.1801	0.2439	0.2532	0.5249	0.5110
LMSE	25	0.0158	0.5375	0.0158	0.5375	0.3637	0.8372
	50	0.0639	0.3772	0.0639	0.4005	0.4210	0.6487
	100	0.0338	0.2467	0.0263	0.3519	0.3981	0.5898
	150	0.0139	0.2260	0.0238	0.3007	0.4402	0.5703
	200	0.0094	0.1877	0.0090	0.2689	0.4483	0.5598
LTSE	25	0.0196	0.3907	0.0196	0.3907	0.3810	0.6326
	50	0.0955	0.2727	0.0955	0.2727	0.4372	0.5599
	100	0.0363	0.1806	0.0601	0.2048	0.4448	0.5300
	150	0.0350	0.1508	0.0514	0.1539	0.4439	0.5156
	200	0.0371	0.1358	0.0622	0.1487	0.4658	0.5078
SE	25	0.0059	0.4191	0.0059	0.4191	0.4574	0.7257
	50	0.0652	0.2859	0.0652	0.2859	0.4620	0.6250
	100	0.0260	0.1775	0.0299	0.2056	0.4497	0.5705
	150	0.0204	0.1548	0.0333	0.1542	0.4946	0.5577
	200	0.0140	0.1294	0.0365	0.1453	0.5012	0.5429
JSE	25	0.4444	0.6128	0.4444	0.6128	0.7679	0.8096
	50	0.4666	0.5501	0.4666	0.5501	0.7722	0.7589
	100	0.3449	0.4417	0.4348	0.5021	0.7062	0.7350
	150	0.3477	0.4209	0.4115	0.4684	0.6812	0.7003
	200	0.3409	0.4020	0.4128	0.4109	0.6663	0.6593
RSE	25	0.2650	0.5878	0.2650	0.5782	0.5192	0.7151
	50	0.3363	0.4982	0.3363	0.5024	0.5473	0.6823
	100	0.2608	0.4054	0.3427	0.4822	0.5167	0.6389
	150	0.2751	0.3947	0.3426	0.4577	0.5207	0.6011
	200	0.2826	0.3772	0.3566	0.4104	0.5291	0.5800

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A10. Estimated Bias and RMSE for ${\epsilon }_i\sim t_2$ with Outliers in the x Direction and no Multicollinearity.

Method	$\mathit{n}$	$\mathit{\pi }=0.10$		$\mathit{\pi }=0.25$		$\mathit{\pi }=0.50$
		Bias	RMSE	Bias	RMSE	Bias	RMSE
OLSE	25	0.2460	0.6138	0.2460	0.8977	0.5062	0.7285
	50	0.3284	0.5014	0.3284	0.6730	0.5376	0.6899
	100	0.2551	0.4067	0.3381	0.5003	0.5133	0.6504
	150	0.2546	0.4000	0.3364	0.4804	0.5158	0.6086
	200	0.2783	0.3824	0.3544	0.4236	0.5260	0.5897
ME	25	0.2329	0.4643	0.2329	0.4745	0.4999	0.6688
	50	0.2970	0.4031	0.2970	0.4299	0.5360	0.6262
	100	0.2078	0.2849	0.2898	0.3909	0.5190	0.5897
	150	0.2336	0.2798	0.2886	0.3667	0.5216	0.5601
	200	0.2234	0.2682	0.3060	0.3500	0.5271	0.5455
MME	25	0.1196	0.4011	0.1196	0.4299	0.4886	0.6631
	50	0.1961	0.3470	0.1961	0.3902	0.5335	0.6092
	100	0.1135	0.2262	0.1969	0.3339	0.5159	0.5383
	150	0.1231	0.2060	0.2112	0.2945	0.5205	0.5175
	200	0.1286	0.1801	0.2439	0.2532	0.5249	0.5110
LMSE	25	0.0158	0.5375	0.0158	0.5375	0.3637	0.8372
	50	0.0639	0.3772	0.0639	0.4005	0.4210	0.6487
	100	0.0338	0.2467	0.0263	0.3519	0.3981	0.5898
	150	0.0139	0.2260	0.0238	0.3007	0.4402	0.5703
	200	0.0094	0.1877	0.0090	0.2689	0.4483	0.5598
LTSE	25	0.0196	0.3907	0.0196	0.3907	0.3810	0.6326
	50	0.0955	0.2727	0.0955	0.2727	0.4372	0.5599
	100	0.0363	0.1806	0.0601	0.2048	0.4448	0.5300
	150	0.0350	0.1508	0.0514	0.1539	0.4439	0.5156
	200	0.0371	0.1358	0.0622	0.1487	0.4658	0.5078
SE	25	0.0059	0.4191	0.0059	0.4191	0.4574	0.7257
	50	0.0652	0.2859	0.0652	0.2859	0.4620	0.6250
	100	0.0260	0.1775	0.0299	0.2056	0.4497	0.5705
	150	0.0204	0.1548	0.0333	0.1542	0.4946	0.5577
	200	0.0140	0.1294	0.0365	0.1453	0.5012	0.5429
JSE	25	0.4444	0.6128	0.4444	0.6128	0.7679	0.8096
	50	0.4666	0.5501	0.4666	0.5501	0.7722	0.7589
	100	0.3449	0.4417	0.4348	0.5021	0.7062	0.7350
	150	0.3477	0.4209	0.4115	0.4684	0.6812	0.7003
	200	0.3409	0.4020	0.4128	0.4109	0.6663	0.6593
RSE	25	0.2650	0.5878	0.2650	0.5782	0.5192	0.7151
	50	0.3363	0.4982	0.3363	0.5024	0.5473	0.6823
	100	0.2608	0.4054	0.3427	0.4822	0.5167	0.6389
	150	0.2751	0.3947	0.3426	0.4577	0.5207	0.6011
	200	0.2826	0.3772	0.3566	0.4104	0.5291	0.5800

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A11. Estimated Bias and RMSE for Normal Errors with 10% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0161	1.1661	0.0267	1.1767	0.0859	4.8447
	50	0.0104	0.9019	0.0146	0.9395	0.0823	3.5972
	100	0.0102	0.6200	0.0049	0.6164	0.0264	2.5442
	150	0.0075	0.5013	0.0039	0.5253	0.0245	2.0862
	200	0.0066	0.4414	0.0045	0.4400	0.0096	1.8097
HME	25	0.0089	0.6654	0.0067	0.6856	0.0170	2.7112
	50	0.0028	0.4462	0.0069	0.4787	0.0182	1.8527
	100	0.0038	0.3104	0.0003	0.3135	0.0257	1.2647
	150	0.0012	0.2425	0.0031	0.2661	0.0156	1.0221
	200	0.0024	0.2164	0.0047	0.2285	0.0083	0.8758
MME	25	0.0022	0.6513	0.0054	0.6630	0.0297	2.5821
	50	0.0007	0.4161	0.0023	0.4474	0.0148	1.7323
	100	0.0019	0.2932	0.0019	0.3061	0.0174	1.2065
	150	0.0023	0.2304	0.0052	0.2452	0.0072	0.9542
	200	0.0024	0.2049	0.0013	0.2196	0.0071	0.8440
LMSE	25	0.0042	1.3739	0.0186	1.4445	0.0502	5.6464
	50	0.0048	0.9668	0.0129	1.0399	0.0760	4.1230
	100	0.0069	0.7582	0.0155	0.8240	0.0185	3.0788
	150	0.0091	0.6626	0.0094	0.6933	0.0459	2.7251
	200	0.0060	0.5967	0.0010	0.6350	0.0252	2.4792
LTSE	25	0.0071	0.8149	0.0092	0.8652	0.0275	3.3599
	50	0.0046	0.4933	0.0080	0.5221	0.0095	2.0934
	100	0.0010	0.3309	0.0036	0.3435	0.0076	1.3464
	150	0.0052	0.2572	0.0015	0.2701	0.0092	1.0902
	200	0.0041	0.2228	0.0002	0.2390	0.0086	0.9219
SE	25	0.0065	1.0166	0.0091	1.0819	0.0270	4.2438
	50	0.0118	0.7430	0.0059	0.7808	0.0462	3.0566
	100	0.0035	0.5346	0.0047	0.5755	0.0378	2.1820
	150	0.0069	0.4442	0.0062	0.4766	0.0322	1.8690
	200	0.0060	0.3978	0.0060	0.4206	0.0118	1.6512
JSE	25	0.0705	1.0623	0.0712	1.4445	0.0855	4.7029
	50	0.0258	0.8135	0.0322	0.8125	0.1237	3.4315
	100	0.0101	0.5772	0.0303	0.5580	0.0744	2.3470
	150	0.0058	0.4759	0.0288	0.4987	0.0558	1.8779
	200	0.0020	0.4091	0.0119	0.4366	0.0604	1.5931
RSE	25	0.0083	0.6170	0.0226	0.5898	0.0942	2.5164
	50	0.0079	0.4132	0.0090	0.4003	0.0612	1.6407
	100	0.0033	0.2935	0.0051	0.2657	0.0500	1.0493
	150	0.0019	0.2310	0.0018	0.2155	0.0323	0.8170
	200	0.0029	0.2015	0.0015	0.1936	0.0359	0.6839

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A11. Estimated Bias and RMSE for Normal Errors with 10% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0161	1.1661	0.0267	1.1767	0.0859	4.8447
	50	0.0104	0.9019	0.0146	0.9395	0.0823	3.5972
	100	0.0102	0.6200	0.0049	0.6164	0.0264	2.5442
	150	0.0075	0.5013	0.0039	0.5253	0.0245	2.0862
	200	0.0066	0.4414	0.0045	0.4400	0.0096	1.8097
HME	25	0.0089	0.6654	0.0067	0.6856	0.0170	2.7112
	50	0.0028	0.4462	0.0069	0.4787	0.0182	1.8527
	100	0.0038	0.3104	0.0003	0.3135	0.0257	1.2647
	150	0.0012	0.2425	0.0031	0.2661	0.0156	1.0221
	200	0.0024	0.2164	0.0047	0.2285	0.0083	0.8758
MME	25	0.0022	0.6513	0.0054	0.6630	0.0297	2.5821
	50	0.0007	0.4161	0.0023	0.4474	0.0148	1.7323
	100	0.0019	0.2932	0.0019	0.3061	0.0174	1.2065
	150	0.0023	0.2304	0.0052	0.2452	0.0072	0.9542
	200	0.0024	0.2049	0.0013	0.2196	0.0071	0.8440
LMSE	25	0.0042	1.3739	0.0186	1.4445	0.0502	5.6464
	50	0.0048	0.9668	0.0129	1.0399	0.0760	4.1230
	100	0.0069	0.7582	0.0155	0.8240	0.0185	3.0788
	150	0.0091	0.6626	0.0094	0.6933	0.0459	2.7251
	200	0.0060	0.5967	0.0010	0.6350	0.0252	2.4792
LTSE	25	0.0071	0.8149	0.0092	0.8652	0.0275	3.3599
	50	0.0046	0.4933	0.0080	0.5221	0.0095	2.0934
	100	0.0010	0.3309	0.0036	0.3435	0.0076	1.3464
	150	0.0052	0.2572	0.0015	0.2701	0.0092	1.0902
	200	0.0041	0.2228	0.0002	0.2390	0.0086	0.9219
SE	25	0.0065	1.0166	0.0091	1.0819	0.0270	4.2438
	50	0.0118	0.7430	0.0059	0.7808	0.0462	3.0566
	100	0.0035	0.5346	0.0047	0.5755	0.0378	2.1820
	150	0.0069	0.4442	0.0062	0.4766	0.0322	1.8690
	200	0.0060	0.3978	0.0060	0.4206	0.0118	1.6512
JSE	25	0.0705	1.0623	0.0712	1.4445	0.0855	4.7029
	50	0.0258	0.8135	0.0322	0.8125	0.1237	3.4315
	100	0.0101	0.5772	0.0303	0.5580	0.0744	2.3470
	150	0.0058	0.4759	0.0288	0.4987	0.0558	1.8779
	200	0.0020	0.4091	0.0119	0.4366	0.0604	1.5931
RSE	25	0.0083	0.6170	0.0226	0.5898	0.0942	2.5164
	50	0.0079	0.4132	0.0090	0.4003	0.0612	1.6407
	100	0.0033	0.2935	0.0051	0.2657	0.0500	1.0493
	150	0.0019	0.2310	0.0018	0.2155	0.0323	0.8170
	200	0.0029	0.2015	0.0015	0.1936	0.0359	0.6839

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A12. Estimated Bias and RMSE Values for Normal Errors with 25% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0062	1.5543	0.0046	1.6247	0.0530	6.3331
	50	0.0212	1.0810	0.0089	1.1396	0.0191	4.4080
	100	0.0137	0.7174	0.0154	0.7447	0.0368	2.8907
	150	0.0038	0.5801	0.0062	0.6160	0.0452	2.4116
	200	0.0057	0.5121	0.0029	0.5368	0.0291	2.1021
HME	25	0.0100	0.7930	0.0035	0.8190	0.0139	3.1968
	50	0.0021	0.4917	0.0063	0.5162	0.0092	1.9931
	100	0.0030	0.3281	0.0013	0.3415	0.0247	1.3352
	150	0.0030	0.2676	0.0023	0.2830	0.0123	1.1082
	200	0.0057	0.2380	0.0056	0.2501	0.0191	0.9796
MME	25	0.0064	0.7174	0.0071	0.7441	0.0280	2.8945
	50	0.0020	0.4460	0.0070	0.4675	0.0163	1.8181
	100	0.0019	0.3435	0.0046	0.3183	0.1093	1.2491
	150	0.0027	0.2773	0.0022	0.2609	0.0150	1.0299
	200	0.0029	0.2187	0.0024	0.2308	0.0145	0.9129
LMSE	25	0.0218	1.4472	0.0095	1.4409	0.0370	5.9381
	50	0.0122	1.0082	0.0089	1.0629	0.0320	4.0632
	100	0.0108	0.7533	0.0069	0.7914	0.0285	2.9913
	150	0.0034	0.6561	0.0024	0.7027	0.0321	2.7053
	200	0.0046	0.6061	0.0042	0.6359	0.0222	2.4734
LTSE	25	0.0062	0.8854	0.0060	0.9104	0.0131	3.5050
	50	0.0041	0.5026	0.0037	0.5218	0.0206	2.0321
	100	0.0026	0.3326	0.0022	0.3432	0.0239	1.3321
	150	0.0045	0.2677	0.0018	0.2782	0.0096	1.0861
	200	0.0022	0.2301	0.0012	0.2423	0.0055	0.9453
SE	25	0.0127	1.0989	0.0099	1.0978	0.0512	4.3564
	50	0.0048	0.8401	0.0057	0.7404	0.0092	2.8994
	100	0.0095	0.6323	0.0018	0.5428	0.0240	2.1071
	150	0.0066	0.5142	0.0052	0.4565	0.0158	1.7641
	200	0.0040	0.4505	0.0034	0.3906	0.0154	1.5235
JSE	25	0.0786	1.3898	0.0726	1.4474	0.0525	6.2080
	50	0.0515	0.9745	0.0477	1.0007	0.1046	4.2547
	100	0.0166	0.6572	0.0200	0.6375	0.1093	2.6981
	150	0.0143	0.5358	0.0192	0.5287	0.1118	2.2113
	200	0.0077	0.4776	0.0126	0.4550	0.0670	1.8900
RSE	25	0.0086	0.6409	0.0226	0.7031	0.0942	3.0150
	50	0.0079	0.4270	0.0090	0.4487	0.0612	1.7837
	100	0.0033	0.3202	0.0051	0.2902	0.0500	1.1195
	150	0.0019	0.2411	0.0018	0.2385	0.0323	0.9004
	200	0.0029	0.1902	0.0015	0.2102	0.0359	0.7816

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A12. Estimated Bias and RMSE Values for Normal Errors with 25% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0062	1.5543	0.0046	1.6247	0.0530	6.3331
	50	0.0212	1.0810	0.0089	1.1396	0.0191	4.4080
	100	0.0137	0.7174	0.0154	0.7447	0.0368	2.8907
	150	0.0038	0.5801	0.0062	0.6160	0.0452	2.4116
	200	0.0057	0.5121	0.0029	0.5368	0.0291	2.1021
HME	25	0.0100	0.7930	0.0035	0.8190	0.0139	3.1968
	50	0.0021	0.4917	0.0063	0.5162	0.0092	1.9931
	100	0.0030	0.3281	0.0013	0.3415	0.0247	1.3352
	150	0.0030	0.2676	0.0023	0.2830	0.0123	1.1082
	200	0.0057	0.2380	0.0056	0.2501	0.0191	0.9796
MME	25	0.0064	0.7174	0.0071	0.7441	0.0280	2.8945
	50	0.0020	0.4460	0.0070	0.4675	0.0163	1.8181
	100	0.0019	0.3435	0.0046	0.3183	0.1093	1.2491
	150	0.0027	0.2773	0.0022	0.2609	0.0150	1.0299
	200	0.0029	0.2187	0.0024	0.2308	0.0145	0.9129
LMSE	25	0.0218	1.4472	0.0095	1.4409	0.0370	5.9381
	50	0.0122	1.0082	0.0089	1.0629	0.0320	4.0632
	100	0.0108	0.7533	0.0069	0.7914	0.0285	2.9913
	150	0.0034	0.6561	0.0024	0.7027	0.0321	2.7053
	200	0.0046	0.6061	0.0042	0.6359	0.0222	2.4734
LTSE	25	0.0062	0.8854	0.0060	0.9104	0.0131	3.5050
	50	0.0041	0.5026	0.0037	0.5218	0.0206	2.0321
	100	0.0026	0.3326	0.0022	0.3432	0.0239	1.3321
	150	0.0045	0.2677	0.0018	0.2782	0.0096	1.0861
	200	0.0022	0.2301	0.0012	0.2423	0.0055	0.9453
SE	25	0.0127	1.0989	0.0099	1.0978	0.0512	4.3564
	50	0.0048	0.8401	0.0057	0.7404	0.0092	2.8994
	100	0.0095	0.6323	0.0018	0.5428	0.0240	2.1071
	150	0.0066	0.5142	0.0052	0.4565	0.0158	1.7641
	200	0.0040	0.4505	0.0034	0.3906	0.0154	1.5235
JSE	25	0.0786	1.3898	0.0726	1.4474	0.0525	6.2080
	50	0.0515	0.9745	0.0477	1.0007	0.1046	4.2547
	100	0.0166	0.6572	0.0200	0.6375	0.1093	2.6981
	150	0.0143	0.5358	0.0192	0.5287	0.1118	2.2113
	200	0.0077	0.4776	0.0126	0.4550	0.0670	1.8900
RSE	25	0.0086	0.6409	0.0226	0.7031	0.0942	3.0150
	50	0.0079	0.4270	0.0090	0.4487	0.0612	1.7837
	100	0.0033	0.3202	0.0051	0.2902	0.0500	1.1195
	150	0.0019	0.2411	0.0018	0.2385	0.0323	0.9004
	200	0.0029	0.1902	0.0015	0.2102	0.0359	0.7816

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A13. Estimated Bias and RMSE Values for Normal Errors with 50% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0361	2.6872	0.0391	3.4565	0.1435	13.4395
	50	0.0207	1.8804	0.0137	2.4060	0.1291	9.2996
	100	0.0229	1.2856	0.0173	1.6108	0.0680	6.2748
	150	0.0054	1.0056	0.0135	1.2991	0.0646	5.0614
	200	0.0152	0.9050	0.0219	1.1438	0.0649	4.4912
HME	25	0.0152	2.0393	0.0385	3.2750	0.2547	12.7506
	50	0.0178	1.2846	0.0166	2.2399	0.0905	8.6701
	100	0.0107	0.8170	0.0108	1.4946	0.0923	5.8517
	150	0.0043	0.6205	0.0192	1.2080	0.1026	4.7199
	200	0.0055	0.5431	0.0188	1.0487	0.0344	4.1304
MME	25	0.0123	2.0160	0.0502	3.3765	0.1710	12.9902
	50	0.0119	1.2511	0.0190	2.2857	0.0689	8.8028
	100	0.0085	0.8068	0.0139	1.5218	0.0902	5.9529
	150	0.0056	0.6260	0.0097	1.2321	0.1193	4.8432
	200	0.0027	0.5310	0.0206	1.0664	0.0504	4.1981
LMSE	25	0.0111	1.8804	0.0700	6.3924	0.2934	24.8944
	50	0.0141	1.1249	0.0352	4.3369	0.1285	16.6505
	100	0.0033	0.7390	0.0336	3.0340	0.0672	11.8651
	150	0.0067	0.6363	0.0238	2.5743	0.0366	9.9337
	200	0.0056	0.5753	0.0365	2.2427	0.0476	8.7208
LTSE	25	0.0126	1.8604	0.0447	4.3584	0.2164	16.9940
	50	0.0069	1.1510	0.0124	2.6902	0.1171	10.4412
	100	0.0074	0.7537	0.0205	1.7371	0.1301	6.8903
	150	0.0089	0.6363	0.0113	1.3817	0.0520	5.3751
	200	0.0050	0.4968	0.0054	1.1846	0.0265	4.6208
SE	25	0.0054	1.5971	0.0354	5.0033	0.1718	18.9508
	50	0.0155	0.9831	0.0254	3.1914	0.0856	12.2507
	100	0.0080	0.6203	0.0418	2.1167	0.1493	8.3920
	150	0.0010	0.5092	0.0101	1.7027	0.0531	6.7311
	200	0.0016	0.4403	0.0184	1.4327	0.0516	5.5473
JSE	25	0.1135	2.4747	0.1148	3.2656	0.1605	13.3709
	50	0.0988	1.6975	0.0892	2.1964	0.0541	9.2018
	100	0.0749	1.1428	0.0778	1.4191	0.0462	6.1440
	150	0.0472	0.9173	0.0748	1.1416	0.0819	4.9121
	200	0.0328	0.8131	0.0387	0.9870	0.0584	4.3278
RSE	25	0.0986	1.5785	0.1026	3.0745	0.2389	12.6750
	50	0.0718	0.8309	0.0875	2.0285	0.0940	8.5669
	100	0.0245	0.4867	0.0564	1.3047	0.0169	5.7115
	150	0.0130	0.3892	0.0506	1.0478	0.1073	4.5634
	200	0.0102	0.3333	0.0408	0.9044	0.0529	3.9618

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A13. Estimated Bias and RMSE Values for Normal Errors with 50% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0361	2.6872	0.0391	3.4565	0.1435	13.4395
	50	0.0207	1.8804	0.0137	2.4060	0.1291	9.2996
	100	0.0229	1.2856	0.0173	1.6108	0.0680	6.2748
	150	0.0054	1.0056	0.0135	1.2991	0.0646	5.0614
	200	0.0152	0.9050	0.0219	1.1438	0.0649	4.4912
HME	25	0.0152	2.0393	0.0385	3.2750	0.2547	12.7506
	50	0.0178	1.2846	0.0166	2.2399	0.0905	8.6701
	100	0.0107	0.8170	0.0108	1.4946	0.0923	5.8517
	150	0.0043	0.6205	0.0192	1.2080	0.1026	4.7199
	200	0.0055	0.5431	0.0188	1.0487	0.0344	4.1304
MME	25	0.0123	2.0160	0.0502	3.3765	0.1710	12.9902
	50	0.0119	1.2511	0.0190	2.2857	0.0689	8.8028
	100	0.0085	0.8068	0.0139	1.5218	0.0902	5.9529
	150	0.0056	0.6260	0.0097	1.2321	0.1193	4.8432
	200	0.0027	0.5310	0.0206	1.0664	0.0504	4.1981
LMSE	25	0.0111	1.8804	0.0700	6.3924	0.2934	24.8944
	50	0.0141	1.1249	0.0352	4.3369	0.1285	16.6505
	100	0.0033	0.7390	0.0336	3.0340	0.0672	11.8651
	150	0.0067	0.6363	0.0238	2.5743	0.0366	9.9337
	200	0.0056	0.5753	0.0365	2.2427	0.0476	8.7208
LTSE	25	0.0126	1.8604	0.0447	4.3584	0.2164	16.9940
	50	0.0069	1.1510	0.0124	2.6902	0.1171	10.4412
	100	0.0074	0.7537	0.0205	1.7371	0.1301	6.8903
	150	0.0089	0.6363	0.0113	1.3817	0.0520	5.3751
	200	0.0050	0.4968	0.0054	1.1846	0.0265	4.6208
SE	25	0.0054	1.5971	0.0354	5.0033	0.1718	18.9508
	50	0.0155	0.9831	0.0254	3.1914	0.0856	12.2507
	100	0.0080	0.6203	0.0418	2.1167	0.1493	8.3920
	150	0.0010	0.5092	0.0101	1.7027	0.0531	6.7311
	200	0.0016	0.4403	0.0184	1.4327	0.0516	5.5473
JSE	25	0.1135	2.4747	0.1148	3.2656	0.1605	13.3709
	50	0.0988	1.6975	0.0892	2.1964	0.0541	9.2018
	100	0.0749	1.1428	0.0778	1.4191	0.0462	6.1440
	150	0.0472	0.9173	0.0748	1.1416	0.0819	4.9121
	200	0.0328	0.8131	0.0387	0.9870	0.0584	4.3278
RSE	25	0.0986	1.5785	0.1026	3.0745	0.2389	12.6750
	50	0.0718	0.8309	0.0875	2.0285	0.0940	8.5669
	100	0.0245	0.4867	0.0564	1.3047	0.0169	5.7115
	150	0.0130	0.3892	0.0506	1.0478	0.1073	4.5634
	200	0.0102	0.3333	0.0408	0.9044	0.0529	3.9618

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A14. Estimated Bias and RMSE Values for Normal Errors with 10% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0135	2.2325	0.0281	2.3509	0.1694	9.0955
	50	0.0263	1.5163	0.0141	1.6087	0.0844	6.2819
	100	0.0101	1.0717	0.0084	1.1340	0.0381	4.4269
	150	0.0078	0.8638	0.0185	0.9072	0.0316	3.5486
	200	0.0127	0.7507	0.0129	0.7909	0.0288	3.0516
HME	25	0.0144	2.1473	0.0092	2.2668	0.0867	8.7886
	50	0.0344	1.4519	0.0249	1.5383	0.0609	5.9849
	100	0.0123	1.0188	0.0082	1.0696	0.0453	4.1529
	150	0.0095	0.8122	0.0089	0.8530	0.0302	3.3366
	200	0.0047	0.7005	0.0093	0.7376	0.0201	2.8468
MME	25	0.0107	2.1610	0.0093	2.2880	0.0801	9.0087
	50	0.0365	1.4752	0.0242	1.5452	0.0408	6.0579
	100	0.0170	1.0358	0.0058	1.0782	0.0595	4.1825
	150	0.0183	0.8209	0.0035	0.8702	0.0559	3.3697
	200	0.0046	0.7107	0.0105	0.7372	0.0416	2.8525
LMSE	25	0.0205	4.5983	0.0626	4.8056	0.2446	19.0631
	50	0.0381	3.2831	0.0553	3.4427	0.1216	13.6482
	100	0.0346	2.3554	0.0554	2.4843	0.1069	9.7381
	150	0.0204	2.0155	0.0385	2.1367	0.0515	8.4898
	200	0.0150	1.8472	0.0211	1.8739	0.0571	7.3646
LTSE	25	0.0272	2.9346	0.0400	3.0534	0.1054	12.2296
	50	0.0189	1.8234	0.0098	1.9280	0.0747	7.6308
	100	0.0194	1.1892	0.0067	1.2630	0.0460	4.8699
	150	0.0064	0.9607	0.0084	1.0068	0.0521	3.9624
	200	0.0043	0.8050	0.0114	0.8567	0.0224	3.2949
SE	25	0.0533	3.4634	0.0468	3.6445	0.1537	14.5047
	50	0.0244	2.4897	0.0347	2.6742	0.0834	10.4343
	100	0.0294	1.7188	0.0229	1.8000	0.0372	6.9285
	150	0.0215	1.4082	0.0176	1.4764	0.0661	5.7540
	200	0.0089	1.1880	0.0051	1.2568	0.0224	4.8324
JSE	25	0.1058	2.0278	0.0689	2.1415	0.0877	8.9953
	50	0.0684	1.3438	0.0585	1.4176	0.0926	6.1550
	100	0.0423	0.9539	0.0499	0.9805	0.0767	4.2635
	150	0.0217	0.7895	0.0265	0.7840	0.1009	3.3695
	200	0.0281	0.6973	0.0348	0.6879	0.0934	2.8600
RSE	25	0.0949	1.9411	0.0778	2.0640	0.0336	8.6892
	50	0.0494	1.2857	0.0664	1.3551	0.0386	5.8501
	100	0.0430	0.9164	0.0547	0.9267	0.0962	3.9854
	150	0.0150	0.7424	0.0255	0.7345	0.1059	3.1533
	200	0.0235	0.6516	0.0381	0.6387	0.0934	2.6459

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A14. Estimated Bias and RMSE Values for Normal Errors with 10% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0135	2.2325	0.0281	2.3509	0.1694	9.0955
	50	0.0263	1.5163	0.0141	1.6087	0.0844	6.2819
	100	0.0101	1.0717	0.0084	1.1340	0.0381	4.4269
	150	0.0078	0.8638	0.0185	0.9072	0.0316	3.5486
	200	0.0127	0.7507	0.0129	0.7909	0.0288	3.0516
HME	25	0.0144	2.1473	0.0092	2.2668	0.0867	8.7886
	50	0.0344	1.4519	0.0249	1.5383	0.0609	5.9849
	100	0.0123	1.0188	0.0082	1.0696	0.0453	4.1529
	150	0.0095	0.8122	0.0089	0.8530	0.0302	3.3366
	200	0.0047	0.7005	0.0093	0.7376	0.0201	2.8468
MME	25	0.0107	2.1610	0.0093	2.2880	0.0801	9.0087
	50	0.0365	1.4752	0.0242	1.5452	0.0408	6.0579
	100	0.0170	1.0358	0.0058	1.0782	0.0595	4.1825
	150	0.0183	0.8209	0.0035	0.8702	0.0559	3.3697
	200	0.0046	0.7107	0.0105	0.7372	0.0416	2.8525
LMSE	25	0.0205	4.5983	0.0626	4.8056	0.2446	19.0631
	50	0.0381	3.2831	0.0553	3.4427	0.1216	13.6482
	100	0.0346	2.3554	0.0554	2.4843	0.1069	9.7381
	150	0.0204	2.0155	0.0385	2.1367	0.0515	8.4898
	200	0.0150	1.8472	0.0211	1.8739	0.0571	7.3646
LTSE	25	0.0272	2.9346	0.0400	3.0534	0.1054	12.2296
	50	0.0189	1.8234	0.0098	1.9280	0.0747	7.6308
	100	0.0194	1.1892	0.0067	1.2630	0.0460	4.8699
	150	0.0064	0.9607	0.0084	1.0068	0.0521	3.9624
	200	0.0043	0.8050	0.0114	0.8567	0.0224	3.2949
SE	25	0.0533	3.4634	0.0468	3.6445	0.1537	14.5047
	50	0.0244	2.4897	0.0347	2.6742	0.0834	10.4343
	100	0.0294	1.7188	0.0229	1.8000	0.0372	6.9285
	150	0.0215	1.4082	0.0176	1.4764	0.0661	5.7540
	200	0.0089	1.1880	0.0051	1.2568	0.0224	4.8324
JSE	25	0.1058	2.0278	0.0689	2.1415	0.0877	8.9953
	50	0.0684	1.3438	0.0585	1.4176	0.0926	6.1550
	100	0.0423	0.9539	0.0499	0.9805	0.0767	4.2635
	150	0.0217	0.7895	0.0265	0.7840	0.1009	3.3695
	200	0.0281	0.6973	0.0348	0.6879	0.0934	2.8600
RSE	25	0.0949	1.9411	0.0778	2.0640	0.0336	8.6892
	50	0.0494	1.2857	0.0664	1.3551	0.0386	5.8501
	100	0.0430	0.9164	0.0547	0.9267	0.0962	3.9854
	150	0.0150	0.7424	0.0255	0.7345	0.1059	3.1533
	200	0.0235	0.6516	0.0381	0.6387	0.0934	2.6459

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A15. Estimated Bias and RMSE Values for Normal Errors with 25% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0281	2.4286	0.0247	2.5558	0.0960	9.9821
	50	0.0326	1.6594	0.0078	1.7535	0.0437	6.7796
	100	0.0136	1.1247	0.0146	1.1722	0.0898	4.5734
	150	0.0103	0.9013	0.0171	0.9516	0.0493	3.7040
	200	0.0075	0.7997	0.0081	0.8411	0.0309	3.3035
HME	25	0.0275	2.3096	0.0108	2.4388	0.0361	9.4162
	50	0.0181	1.5310	0.0308	1.6133	0.0503	6.1978
	100	0.0091	1.0456	0.0143	1.0929	0.1120	4.2635
	150	0.0118	0.8415	0.0064	0.8870	0.0390	3.4689
	200	0.0066	0.7483	0.0098	0.7872	0.0170	3.0805
MME	25	0.0349	2.3528	0.0125	2.4749	0.0507	9.5822
	50	0.0203	1.5405	0.0272	1.6325	0.0503	6.2386
	100	0.0157	1.0569	0.0211	1.1076	0.1020	4.3226
	150	0.0059	0.8486	0.0060	0.8878	0.0290	3.5174
	200	0.0066	0.7566	0.0126	0.7982	0.0120	3.1332
LMSE	25	0.0683	4.9767	0.0417	5.2389	0.1866	19.8633
	50	0.0586	3.4244	0.0647	3.5244	0.3311	13.4509
	100	0.0313	2.4225	0.0572	2.5611	0.0303	9.6458
	150	0.0295	2.0941	0.0110	2.1461	0.0874	8.4162
	200	0.0024	1.8611	0.0328	1.9605	0.0901	7.6914
LTSE	25	0.0260	3.2252	0.0309	3.4264	0.1217	13.2600
	50	0.0071	1.8757	0.0271	1.9788	0.1588	7.5153
	100	0.0185	1.2288	0.0070	1.2818	0.0851	5.0267
	150	0.0063	0.9946	0.0084	1.0384	0.0529	4.0800
	200	0.0094	0.8717	0.0096	0.9133	0.0515	3.5295
SE	25	0.0354	3.7383	0.0530	4.0627	0.0944	15.9112
	50	0.0174	2.4830	0.0605	2.6136	0.2457	10.1470
	100	0.0102	1.7304	0.0191	1.8229	0.0568	7.1198
	150	0.0236	1.3918	0.0101	1.4741	0.0348	5.6401
	200	0.0185	1.2410	0.0301	1.3127	0.0663	5.0493
JSE	25	0.0838	2.2172	0.1500	2.3635	0.0430	9.8877
	50	0.0962	1.4831	0.0885	1.5707	0.0312	6.0673
	100	0.0434	0.9977	0.0552	1.0160	0.1486	4.4244
	150	0.0290	0.8187	0.0299	0.8197	0.0966	3.5270
	200	0.0191	0.7291	0.0371	0.7239	0.0727	3.1188
RSE	25	0.0900	2.1028	0.0717	2.2381	0.0566	9.3223
	50	0.0846	1.3616	0.0952	1.4390	0.1143	6.0673
	100	0.0521	0.9346	0.0482	0.9475	0.1295	4.1004
	150	0.0289	0.7667	0.0304	0.7642	0.0856	3.2875
	200	0.0163	0.6869	0.0270	0.6772	0.0723	2.8904

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A15. Estimated Bias and RMSE Values for Normal Errors with 25% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0281	2.4286	0.0247	2.5558	0.0960	9.9821
	50	0.0326	1.6594	0.0078	1.7535	0.0437	6.7796
	100	0.0136	1.1247	0.0146	1.1722	0.0898	4.5734
	150	0.0103	0.9013	0.0171	0.9516	0.0493	3.7040
	200	0.0075	0.7997	0.0081	0.8411	0.0309	3.3035
HME	25	0.0275	2.3096	0.0108	2.4388	0.0361	9.4162
	50	0.0181	1.5310	0.0308	1.6133	0.0503	6.1978
	100	0.0091	1.0456	0.0143	1.0929	0.1120	4.2635
	150	0.0118	0.8415	0.0064	0.8870	0.0390	3.4689
	200	0.0066	0.7483	0.0098	0.7872	0.0170	3.0805
MME	25	0.0349	2.3528	0.0125	2.4749	0.0507	9.5822
	50	0.0203	1.5405	0.0272	1.6325	0.0503	6.2386
	100	0.0157	1.0569	0.0211	1.1076	0.1020	4.3226
	150	0.0059	0.8486	0.0060	0.8878	0.0290	3.5174
	200	0.0066	0.7566	0.0126	0.7982	0.0120	3.1332
LMSE	25	0.0683	4.9767	0.0417	5.2389	0.1866	19.8633
	50	0.0586	3.4244	0.0647	3.5244	0.3311	13.4509
	100	0.0313	2.4225	0.0572	2.5611	0.0303	9.6458
	150	0.0295	2.0941	0.0110	2.1461	0.0874	8.4162
	200	0.0024	1.8611	0.0328	1.9605	0.0901	7.6914
LTSE	25	0.0260	3.2252	0.0309	3.4264	0.1217	13.2600
	50	0.0071	1.8757	0.0271	1.9788	0.1588	7.5153
	100	0.0185	1.2288	0.0070	1.2818	0.0851	5.0267
	150	0.0063	0.9946	0.0084	1.0384	0.0529	4.0800
	200	0.0094	0.8717	0.0096	0.9133	0.0515	3.5295
SE	25	0.0354	3.7383	0.0530	4.0627	0.0944	15.9112
	50	0.0174	2.4830	0.0605	2.6136	0.2457	10.1470
	100	0.0102	1.7304	0.0191	1.8229	0.0568	7.1198
	150	0.0236	1.3918	0.0101	1.4741	0.0348	5.6401
	200	0.0185	1.2410	0.0301	1.3127	0.0663	5.0493
JSE	25	0.0838	2.2172	0.1500	2.3635	0.0430	9.8877
	50	0.0962	1.4831	0.0885	1.5707	0.0312	6.0673
	100	0.0434	0.9977	0.0552	1.0160	0.1486	4.4244
	150	0.0290	0.8187	0.0299	0.8197	0.0966	3.5270
	200	0.0191	0.7291	0.0371	0.7239	0.0727	3.1188
RSE	25	0.0900	2.1028	0.0717	2.2381	0.0566	9.3223
	50	0.0846	1.3616	0.0952	1.4390	0.1143	6.0673
	100	0.0521	0.9346	0.0482	0.9475	0.1295	4.1004
	150	0.0289	0.7667	0.0304	0.7642	0.0856	3.2875
	200	0.0163	0.6869	0.0270	0.6772	0.0723	2.8904

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A16. Estimated Bias and RMSE Values for Normal Errors with 50% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0257	3.2900	0.0391	3.4565	0.1435	13.4395
	50	0.0204	2.2706	0.0137	2.4060	0.1291	9.2996
	100	0.0120	1.5474	0.0173	1.6108	0.0680	6.2748
	150	0.0127	1.2354	0.0135	1.2991	0.0646	5.0614
	200	0.0123	1.0907	0.0219	1.1438	0.0649	4.4912
HME	25	0.0311	3.1178	0.0385	3.2750	0.2547	12.7506
	50	0.0172	2.1034	0.0166	2.2399	0.0905	8.6701
	100	0.0211	1.4313	0.0108	1.4946	0.0923	5.8517
	150	0.0149	1.1512	0.0192	1.2080	0.1026	4.7199
	200	0.0244	1.0038	0.0188	1.0487	0.0344	4.1304
MME	25	0.0491	3.1932	0.0502	3.3765	0.1710	12.9902
	50	0.0248	2.1431	0.0190	2.2857	0.0689	8.8028
	100	0.0204	1.4502	0.0139	1.5218	0.0902	5.9529
	150	0.0165	1.1734	0.0097	1.2321	0.1193	4.8432
	200	0.0244	1.0204	0.0206	1.0664	0.0504	4.1981
LMSE	25	0.0645	6.0669	0.0700	6.3924	0.2934	24.8944
	50	0.0437	4.1302	0.0352	4.3695	0.1285	16.6505
	100	0.0353	2.8316	0.0336	3.0340	0.0672	11.8651
	150	0.0203	2.4174	0.0238	2.5743	0.0366	9.9337
	200	0.0136	2.1420	0.0337	2.2427	0.0476	8.7208
LTSE	25	0.0761	4.0590	0.0457	4.3584	0.2164	16.9940
	50	0.0180	2.5061	0.0124	2.6902	0.1171	10.4412
	100	0.0214	1.6536	0.0205	1.7371	0.1301	6.8903
	150	0.0062	1.3165	0.0113	1.3817	0.0520	6.7311
	200	0.0110	1.1377	0.0054	1.1846	0.0265	5.5473
SE	25	0.0337	4.5980	0.0354	5.0033	0.1718	18.9508
	50	0.0206	3.0280	0.0254	3.1914	0.0856	12.2507
	100	0.0307	2.0286	0.0418	2.1167	0.1493	8.3920
	150	0.0246	1.6297	0.0101	1.7027	0.0531	6.7311
	200	0.0133	1.3705	0.0184	1.4327	0.0516	4.5473
JSE	25	0.1499	3.0926	0.1148	3.2656	0.1605	13.3709
	50	0.1395	2.0736	0.0875	2.1964	0.0541	9.2018
	100	0.0838	1.3648	0.0778	1.4191	0.0462	6.1440
	150	0.0664	1.1109	0.0748	1.1416	0.0819	4.9121
	200	0.0398	0.9737	0.0387	0.9870	0.0584	4.3278
RSE	25	0.1312	2.9141	0.1026	3.0745	0.2389	12.6750
	50	0.1123	1.8990	0.0875	2.0285	0.0940	8.5669
	100	0.0838	1.2641	0.0564	1.3057	0.0169	5.7115
	150	0.0538	1.0271	0.0506	1.0478	0.1073	4.5634
	200	0.0466	0.9041	0.0408	0.9044	0.0529	3.9618

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A16. Estimated Bias and RMSE Values for Normal Errors with 50% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0257	3.2900	0.0391	3.4565	0.1435	13.4395
	50	0.0204	2.2706	0.0137	2.4060	0.1291	9.2996
	100	0.0120	1.5474	0.0173	1.6108	0.0680	6.2748
	150	0.0127	1.2354	0.0135	1.2991	0.0646	5.0614
	200	0.0123	1.0907	0.0219	1.1438	0.0649	4.4912
HME	25	0.0311	3.1178	0.0385	3.2750	0.2547	12.7506
	50	0.0172	2.1034	0.0166	2.2399	0.0905	8.6701
	100	0.0211	1.4313	0.0108	1.4946	0.0923	5.8517
	150	0.0149	1.1512	0.0192	1.2080	0.1026	4.7199
	200	0.0244	1.0038	0.0188	1.0487	0.0344	4.1304
MME	25	0.0491	3.1932	0.0502	3.3765	0.1710	12.9902
	50	0.0248	2.1431	0.0190	2.2857	0.0689	8.8028
	100	0.0204	1.4502	0.0139	1.5218	0.0902	5.9529
	150	0.0165	1.1734	0.0097	1.2321	0.1193	4.8432
	200	0.0244	1.0204	0.0206	1.0664	0.0504	4.1981
LMSE	25	0.0645	6.0669	0.0700	6.3924	0.2934	24.8944
	50	0.0437	4.1302	0.0352	4.3695	0.1285	16.6505
	100	0.0353	2.8316	0.0336	3.0340	0.0672	11.8651
	150	0.0203	2.4174	0.0238	2.5743	0.0366	9.9337
	200	0.0136	2.1420	0.0337	2.2427	0.0476	8.7208
LTSE	25	0.0761	4.0590	0.0457	4.3584	0.2164	16.9940
	50	0.0180	2.5061	0.0124	2.6902	0.1171	10.4412
	100	0.0214	1.6536	0.0205	1.7371	0.1301	6.8903
	150	0.0062	1.3165	0.0113	1.3817	0.0520	6.7311
	200	0.0110	1.1377	0.0054	1.1846	0.0265	5.5473
SE	25	0.0337	4.5980	0.0354	5.0033	0.1718	18.9508
	50	0.0206	3.0280	0.0254	3.1914	0.0856	12.2507
	100	0.0307	2.0286	0.0418	2.1167	0.1493	8.3920
	150	0.0246	1.6297	0.0101	1.7027	0.0531	6.7311
	200	0.0133	1.3705	0.0184	1.4327	0.0516	4.5473
JSE	25	0.1499	3.0926	0.1148	3.2656	0.1605	13.3709
	50	0.1395	2.0736	0.0875	2.1964	0.0541	9.2018
	100	0.0838	1.3648	0.0778	1.4191	0.0462	6.1440
	150	0.0664	1.1109	0.0748	1.1416	0.0819	4.9121
	200	0.0398	0.9737	0.0387	0.9870	0.0584	4.3278
RSE	25	0.1312	2.9141	0.1026	3.0745	0.2389	12.6750
	50	0.1123	1.8990	0.0875	2.0285	0.0940	8.5669
	100	0.0838	1.2641	0.0564	1.3057	0.0169	5.7115
	150	0.0538	1.0271	0.0506	1.0478	0.1073	4.5634
	200	0.0466	0.9041	0.0408	0.9044	0.0529	3.9618

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A17. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 10% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0355	2.6985	0.0653	2.8377	0.2089	9.2302
	50	0.0320	2.0576	0.0495	2.0770	0.1682	6.7577
	100	0.0432	1.4167	0.0319	1.4718	0.1120	4.6603
	150	0.0193	1.1467	0.0397	1.1803	0.1703	3.7981
	200	0.0354	0.9816	0.0358	1.0025	0.0565	3.1780
HME	25	0.0128	0.7385	0.0191	0.7786	0.0294	2.4807
	50	0.0077	0.4569	0.0275	0.4823	0.0326	1.5343
	100	0.0099	0.3093	0.0030	0.3199	0.0089	1.0333
	150	0.0018	0.2471	0.0038	0.2575	0.0173	0.8176
	200	0.0039	0.2129	0.0094	0.2207	0.0171	0.6987
MME	25	0.0158	0.7469	0.0195	0.7784	0.0311	2.4392
	50	0.0077	0.4638	0.0300	0.4883	0.0375	1.5615
	100	0.0086	0.3129	0.0027	0.3233	0.0065	1.0473
	150	0.0016	0.2493	0.0038	0.2594	0.0190	0.8263
	200	0.0042	0.2152	0.0095	0.2230	0.0167	0.7059
LMSE	25	0.0290	1.1745	0.0626	1.2612	0.1298	4.0206
	50	0.0253	0.7511	0.0316	0.7720	0.1194	2.5813
	100	0.0116	0.4959	0.0168	0.5316	0.0427	1.6684
	150	0.0121	0.4128	0.0056	0.4416	0.0336	1.3965
	200	0.0115	0.3746	0.0096	0.3859	0.0415	1.2230
LTSE	25	0.0252	0.7879	0.0193	0.8346	0.0366	2.6328
	50	0.0072	0.4848	0.0269	0.5010	0.0346	1.6078
	100	0.0142	0.3152	0.0040	0.3235	0.0088	1.0657
	150	0.0012	0.2513	0.0031	0.2666	0.0148	0.8401
	200	0.0043	0.2158	0.0086	0.2244	0.0174	0.7055
SE	25	0.0276	0.8549	0.0235	0.8870	0.0585	2.9387
	50	0.0185	0.5267	0.0266	0.5610	0.0671	1.8034
	100	0.0099	0.3323	0.0085	0.3532	0.0227	1.1202
	150	0.0057	0.2680	0.0035	0.2835	0.0095	0.9089
	200	0.0088	0.2276	0.0086	0.2383	0.0127	0.7596
JSE	25	0.7574	1.3869	0.7617	1.4108	0.7701	1.8679
	50	0.6873	1.2240	0.7396	1.2222	0.7502	1.6907
	100	0.5537	1.0248	0.6130	1.0448	0.7646	1.0864
	150	0.4985	0.9498	0.5308	0.9672	0.7523	1.0285
	200	0.4551	0.8875	0.5165	0.9029	0.7547	0.9957
RSE	25	0.1224	0.7129	0.1233	0.7489	0.2582	1.4354
	50	0.1200	0.4517	0.1200	0.4631	0.2510	1.1576
	100	0.1123	0.3073	0.1130	0.3143	0.2413	0.7914
	150	0.1126	0.2284	0.1132	0.2495	0.2421	0.6433
	200	0.1164	0.2003	0.1159	0.2105	0.2397	0.5571

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A17. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 10% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0355	2.6985	0.0653	2.8377	0.2089	9.2302
	50	0.0320	2.0576	0.0495	2.0770	0.1682	6.7577
	100	0.0432	1.4167	0.0319	1.4718	0.1120	4.6603
	150	0.0193	1.1467	0.0397	1.1803	0.1703	3.7981
	200	0.0354	0.9816	0.0358	1.0025	0.0565	3.1780
HME	25	0.0128	0.7385	0.0191	0.7786	0.0294	2.4807
	50	0.0077	0.4569	0.0275	0.4823	0.0326	1.5343
	100	0.0099	0.3093	0.0030	0.3199	0.0089	1.0333
	150	0.0018	0.2471	0.0038	0.2575	0.0173	0.8176
	200	0.0039	0.2129	0.0094	0.2207	0.0171	0.6987
MME	25	0.0158	0.7469	0.0195	0.7784	0.0311	2.4392
	50	0.0077	0.4638	0.0300	0.4883	0.0375	1.5615
	100	0.0086	0.3129	0.0027	0.3233	0.0065	1.0473
	150	0.0016	0.2493	0.0038	0.2594	0.0190	0.8263
	200	0.0042	0.2152	0.0095	0.2230	0.0167	0.7059
LMSE	25	0.0290	1.1745	0.0626	1.2612	0.1298	4.0206
	50	0.0253	0.7511	0.0316	0.7720	0.1194	2.5813
	100	0.0116	0.4959	0.0168	0.5316	0.0427	1.6684
	150	0.0121	0.4128	0.0056	0.4416	0.0336	1.3965
	200	0.0115	0.3746	0.0096	0.3859	0.0415	1.2230
LTSE	25	0.0252	0.7879	0.0193	0.8346	0.0366	2.6328
	50	0.0072	0.4848	0.0269	0.5010	0.0346	1.6078
	100	0.0142	0.3152	0.0040	0.3235	0.0088	1.0657
	150	0.0012	0.2513	0.0031	0.2666	0.0148	0.8401
	200	0.0043	0.2158	0.0086	0.2244	0.0174	0.7055
SE	25	0.0276	0.8549	0.0235	0.8870	0.0585	2.9387
	50	0.0185	0.5267	0.0266	0.5610	0.0671	1.8034
	100	0.0099	0.3323	0.0085	0.3532	0.0227	1.1202
	150	0.0057	0.2680	0.0035	0.2835	0.0095	0.9089
	200	0.0088	0.2276	0.0086	0.2383	0.0127	0.7596
JSE	25	0.7574	1.3869	0.7617	1.4108	0.7701	1.8679
	50	0.6873	1.2240	0.7396	1.2222	0.7502	1.6907
	100	0.5537	1.0248	0.6130	1.0448	0.7646	1.0864
	150	0.4985	0.9498	0.5308	0.9672	0.7523	1.0285
	200	0.4551	0.8875	0.5165	0.9029	0.7547	0.9957
RSE	25	0.1224	0.7129	0.1233	0.7489	0.2582	1.4354
	50	0.1200	0.4517	0.1200	0.4631	0.2510	1.1576
	100	0.1123	0.3073	0.1130	0.3143	0.2413	0.7914
	150	0.1126	0.2284	0.1132	0.2495	0.2421	0.6433
	200	0.1164	0.2003	0.1159	0.2105	0.2397	0.5571

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A18. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with Multicollinearity and 25% Outliers in the y Direction.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0851	4.4032	0.1041	4.5218	0.4094	14.3599
	50	0.0658	2.9326	0.0482	3.0069	0.3181	9.7475
	100	0.1061	2.1240	0.0376	2.1299	0.3463	7.0300
	150	0.0412	1.6970	0.0381	1.7547	0.1705	5.3059
	200	0.0538	1.4902	0.0549	1.5066	0.1883	4.7391
HME	25	0.0308	1.4874	0.0160	1.5534	0.2438	4.8113
	50	0.0183	0.5697	0.0149	0.5926	0.0435	1.9026
	100	0.0115	0.3552	0.0117	0.3766	0.0294	0.9398
	150	0.0075	0.2912	0.0033	0.2994	0.0294	0.9398
	200	0.0058	0.2434	0.0066	0.2565	0.0235	0.7541
MME	25	0.0157	1.1182	0.0193	1.1224	0.0734	3.6637
	50	0.0157	0.5721	0.0118	0.6094	0.0330	1.9400
	100	0.0122	0.3691	0.0134	0.3909	0.0258	0.9677
	150	0.0080	0.3006	0.0032	0.3098	0.0258	0.9677
	200	0.0055	0.2507	0.0064	0.2638	0.0190	0.8570
LMSE	25	0.0338	1.2257	0.0361	1.2559	0.1089	4.0461
	50	0.0072	0.7788	0.0252	0.8096	0.0647	2.5366
	100	0.0081	0.5321	0.0176	0.5589	0.0429	1.7388
	150	0.0191	0.4592	0.0113	0.4624	0.0448	1.4856
	200	0.0075	0.4087	0.0086	0.4204	0.0120	1.3296
LTSE	25	0.0268	0.8817	0.0122	0.9245	0.0879	2.9881
	50	0.0119	0.5463	0.0140	0.5711	0.0339	1.8242
	100	0.0103	0.3505	0.0096	0.3767	0.0106	1.2141
	150	0.0079	0.2930	0.0054	0.2985	0.0324	0.9495
	200	0.0073	0.2445	0.0066	0.2566	0.0201	0.8194
SE	25	0.0336	0.8569	0.0086	0.8580	0.1036	2.8933
	50	0.0091	0.5331	0.0194	0.5448	0.0678	1.7469
	100	0.0084	0.3461	0.0077	0.3611	0.0160	1.1448
	150	0.0086	0.2966	0.0058	0.2981	0.0428	0.9002
	200	0.0079	0.2547	0.0074	0.2619	0.0235	0.7541
JSE	25	0.7908	1.8380	0.7576	1.7894	0.7684	10.8075
	50	0.7524	1.4694	0.7724	1.4720	0.7594	8.4858
	100	0.7156	1.2321	0.7597	1.2758	0.7571	4.2756
	150	0.6420	1.1254	0.7081	1.1349	0.7642	3.1536
	200	0.5570	1.0665	0.6444	1.0662	0.7670	2.5951
RSE	25	0.1144	0.7906	0.1131	0.8281	0.2498	2.1158
	50	0.1181	0.5053	0.1187	0.5235	0.2424	1.3083
	100	0.1241	0.3315	0.1168	0.3544	0.2384	0.8901
	150	0.1182	0.2685	0.1211	0.2755	0.2412	0.7147
	200	0.1116	0.2276	0.1144	0.2380	0.2428	0.6306

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A18. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with Multicollinearity and 25% Outliers in the y Direction.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.0851	4.4032	0.1041	4.5218	0.4094	14.3599
	50	0.0658	2.9326	0.0482	3.0069	0.3181	9.7475
	100	0.1061	2.1240	0.0376	2.1299	0.3463	7.0300
	150	0.0412	1.6970	0.0381	1.7547	0.1705	5.3059
	200	0.0538	1.4902	0.0549	1.5066	0.1883	4.7391
HME	25	0.0308	1.4874	0.0160	1.5534	0.2438	4.8113
	50	0.0183	0.5697	0.0149	0.5926	0.0435	1.9026
	100	0.0115	0.3552	0.0117	0.3766	0.0294	0.9398
	150	0.0075	0.2912	0.0033	0.2994	0.0294	0.9398
	200	0.0058	0.2434	0.0066	0.2565	0.0235	0.7541
MME	25	0.0157	1.1182	0.0193	1.1224	0.0734	3.6637
	50	0.0157	0.5721	0.0118	0.6094	0.0330	1.9400
	100	0.0122	0.3691	0.0134	0.3909	0.0258	0.9677
	150	0.0080	0.3006	0.0032	0.3098	0.0258	0.9677
	200	0.0055	0.2507	0.0064	0.2638	0.0190	0.8570
LMSE	25	0.0338	1.2257	0.0361	1.2559	0.1089	4.0461
	50	0.0072	0.7788	0.0252	0.8096	0.0647	2.5366
	100	0.0081	0.5321	0.0176	0.5589	0.0429	1.7388
	150	0.0191	0.4592	0.0113	0.4624	0.0448	1.4856
	200	0.0075	0.4087	0.0086	0.4204	0.0120	1.3296
LTSE	25	0.0268	0.8817	0.0122	0.9245	0.0879	2.9881
	50	0.0119	0.5463	0.0140	0.5711	0.0339	1.8242
	100	0.0103	0.3505	0.0096	0.3767	0.0106	1.2141
	150	0.0079	0.2930	0.0054	0.2985	0.0324	0.9495
	200	0.0073	0.2445	0.0066	0.2566	0.0201	0.8194
SE	25	0.0336	0.8569	0.0086	0.8580	0.1036	2.8933
	50	0.0091	0.5331	0.0194	0.5448	0.0678	1.7469
	100	0.0084	0.3461	0.0077	0.3611	0.0160	1.1448
	150	0.0086	0.2966	0.0058	0.2981	0.0428	0.9002
	200	0.0079	0.2547	0.0074	0.2619	0.0235	0.7541
JSE	25	0.7908	1.8380	0.7576	1.7894	0.7684	10.8075
	50	0.7524	1.4694	0.7724	1.4720	0.7594	8.4858
	100	0.7156	1.2321	0.7597	1.2758	0.7571	4.2756
	150	0.6420	1.1254	0.7081	1.1349	0.7642	3.1536
	200	0.5570	1.0665	0.6444	1.0662	0.7670	2.5951
RSE	25	0.1144	0.7906	0.1131	0.8281	0.2498	2.1158
	50	0.1181	0.5053	0.1187	0.5235	0.2424	1.3083
	100	0.1241	0.3315	0.1168	0.3544	0.2384	0.8901
	150	0.1182	0.2685	0.1211	0.2755	0.2412	0.7147
	200	0.1116	0.2276	0.1144	0.2380	0.2428	0.6306

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A19. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 50% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.8488	19.6873	0.7906	19.4488	1.0191	40.7973
	50	0.5346	19.0980	0.6276	19.3425	1.2897	30.2086
	100	0.7694	18.0536	0.7018	18.9754	0.8172	24.9392
	150	0.6870	17.1070	0.4330	18.3462	1.0011	22.1539
	200	0.5887	16.3318	0.4991	17.3047	0.7971	20.7615
HME	25	0.3293	7.0647	0.1116	7.3974	0.4452	24.0466
	50	0.1449	4.7746	0.1662	4.9250	0.5889	15.9805
	100	0.1303	3.2177	0.1267	2.8200	0.1635	11.1029
	150	0.1416	2.6552	0.1113	2.4335	0.1934	8.9369
	200	0.0943	2.3010	0.0537	2.0563	0.1468	7.5880
MME	25	0.2795	6.6162	0.0981	6.8656	0.4049	22.4089
	50	0.1324	4.5829	0.1512	4.7748	0.5941	15.3914
	100	0.1227	3.1681	0.1144	3.3225	0.1632	10.9061
	150	0.1379	2.6262	0.1059	2.7004	0.1905	8.8067
	200	0.0825	2.2941	0.0405	2.3425	0.1613	7.5240
LMSE	25	0.1442	7.4609	0.1380	8.0187	0.3155	18.9403
	50	0.0989	5.1311	0.1235	5.3363	0.2125	16.5584
	100	0.0634	3.2493	0.1848	3.3769	0.1479	11.0435
	150	0.0803	2.6496	0.0792	3.0367	0.0967	8.8498
	200	0.1008	2.6153	0.0984	2.6756	0.2632	7.7167
LTSE	25	0.4722	8.3560	0.1881	8.9986	0.7007	28.8922
	50	0.1518	5.4782	0.2335	5.7422	0.7990	18.5412
	100	0.1381	3.2822	0.1341	3.4588	0.2280	11.3549
	150	0.1360	2.6971	0.1302	2.7673	0.1870	9.1910
	200	0.1052	2.3156	0.0656	2.3916	0.1188	7.7337
SE	25	2.8435	9.4556	3.7004	10.4928	5.4429	28.2871
	50	0.8114	6.5779	1.2565	6.2095	2.0388	16.3748
	100	0.6504	3.8305	0.9445	3.9316	1.9139	11.8926
	150	0.9939	2.7156	0.7543	3.2098	5.5093	9.5507
	200	0.8905	2.4521	0.5439	2.8587	3.0965	8.8487
JSE	25	0.7432	19.5232	0.6327	19.1775	0.9350	30.9780
	50	0.5909	18.0380	0.9221	18.4486	0.3600	27.1521
	100	0.4665	17.0666	0.4645	18.0012	0.5687	19.1521
	150	0.4083	16.0001	0.7556	17.3483	1.1434	18.1797
	200	0.5706	15.0623	0.6031	16.2746	0.5111	17.1067
RSE	25	0.3469	4.5985	0.1614	4.5109	0.1272	13.7179
	50	0.0791	3.1763	0.1239	3.2625	0.7092	9.8212
	100	0.0890	2.3176	0.0884	2.5966	0.1543	7.8820
	150	0.1325	2.0001	0.0942	2.1660	0.1484	7.0053
	200	0.0768	1.8924	0.0263	1.8990	0.1324	6.4548

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A19. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 50% Outliers in the y Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.8488	19.6873	0.7906	19.4488	1.0191	40.7973
	50	0.5346	19.0980	0.6276	19.3425	1.2897	30.2086
	100	0.7694	18.0536	0.7018	18.9754	0.8172	24.9392
	150	0.6870	17.1070	0.4330	18.3462	1.0011	22.1539
	200	0.5887	16.3318	0.4991	17.3047	0.7971	20.7615
HME	25	0.3293	7.0647	0.1116	7.3974	0.4452	24.0466
	50	0.1449	4.7746	0.1662	4.9250	0.5889	15.9805
	100	0.1303	3.2177	0.1267	2.8200	0.1635	11.1029
	150	0.1416	2.6552	0.1113	2.4335	0.1934	8.9369
	200	0.0943	2.3010	0.0537	2.0563	0.1468	7.5880
MME	25	0.2795	6.6162	0.0981	6.8656	0.4049	22.4089
	50	0.1324	4.5829	0.1512	4.7748	0.5941	15.3914
	100	0.1227	3.1681	0.1144	3.3225	0.1632	10.9061
	150	0.1379	2.6262	0.1059	2.7004	0.1905	8.8067
	200	0.0825	2.2941	0.0405	2.3425	0.1613	7.5240
LMSE	25	0.1442	7.4609	0.1380	8.0187	0.3155	18.9403
	50	0.0989	5.1311	0.1235	5.3363	0.2125	16.5584
	100	0.0634	3.2493	0.1848	3.3769	0.1479	11.0435
	150	0.0803	2.6496	0.0792	3.0367	0.0967	8.8498
	200	0.1008	2.6153	0.0984	2.6756	0.2632	7.7167
LTSE	25	0.4722	8.3560	0.1881	8.9986	0.7007	28.8922
	50	0.1518	5.4782	0.2335	5.7422	0.7990	18.5412
	100	0.1381	3.2822	0.1341	3.4588	0.2280	11.3549
	150	0.1360	2.6971	0.1302	2.7673	0.1870	9.1910
	200	0.1052	2.3156	0.0656	2.3916	0.1188	7.7337
SE	25	2.8435	9.4556	3.7004	10.4928	5.4429	28.2871
	50	0.8114	6.5779	1.2565	6.2095	2.0388	16.3748
	100	0.6504	3.8305	0.9445	3.9316	1.9139	11.8926
	150	0.9939	2.7156	0.7543	3.2098	5.5093	9.5507
	200	0.8905	2.4521	0.5439	2.8587	3.0965	8.8487
JSE	25	0.7432	19.5232	0.6327	19.1775	0.9350	30.9780
	50	0.5909	18.0380	0.9221	18.4486	0.3600	27.1521
	100	0.4665	17.0666	0.4645	18.0012	0.5687	19.1521
	150	0.4083	16.0001	0.7556	17.3483	1.1434	18.1797
	200	0.5706	15.0623	0.6031	16.2746	0.5111	17.1067
RSE	25	0.3469	4.5985	0.1614	4.5109	0.1272	13.7179
	50	0.0791	3.1763	0.1239	3.2625	0.7092	9.8212
	100	0.0890	2.3176	0.0884	2.5966	0.1543	7.8820
	150	0.1325	2.0001	0.0942	2.1660	0.1484	7.0053
	200	0.0768	1.8924	0.0263	1.8990	0.1324	6.4548

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A20. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 10% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.7944	5.2375	0.5017	6.1350	0.6275	6.4083
	50	0.5521	4.8550	0.7120	5.9626	0.4389	6.2877
	100	0.8544	4.1828	0.4655	5.7585	0.6999	6.0153
	150	0.7160	3.0651	0.5376	5.3560	0.5693	5.2649
	200	0.5921	2.9390	0.5933	5.2213	0.6371	5.0758
HME	25	0.0695	0.5643	0.0608	0.5857	0.0679	0.8973
	50	0.0690	0.3065	0.0660	0.2964	0.0626	0.3229
	100	0.0623	0.2039	0.0598	0.2038	0.0604	0.2116
	150	0.0600	0.1724	0.0624	0.1757	0.0629	0.1757
	200	0.0614	0.1599	0.0619	0.1614	0.0601	0.1588
MME	25	0.0562	0.5725	0.0457	0.5904	0.0489	1.2000
	50	0.0562	0.2944	0.0548	0.2909	0.0484	0.3485
	100	0.0468	0.1900	0.0464	0.1906	0.0450	0.1981
	150	0.0458	0.1560	0.0484	0.1639	0.0481	0.1609
	200	0.0463	0.1433	0.0448	0.1420	0.0452	0.1447
LMSE	25	0.0625	1.1280	0.0498	1.1404	0.0931	3.3462
	50	0.0519	0.6491	0.0658	0.6595	0.0648	1.3806
	100	0.0548	0.4080	0.0576	0.4131	0.0598	0.5696
	150	0.0529	0.3264	0.0605	0.3239	0.0503	0.3914
	200	0.0566	0.2774	0.0519	0.2713	0.0580	0.3142
LTSE	25	0.0973	0.6964	0.0647	0.6953	0.0736	1.6717
	50	0.0780	0.3913	0.0783	0.3967	0.0724	0.6050
	100	0.0750	0.2562	0.0680	0.2529	0.0700	0.3075
	150	0.0673	0.2046	0.0731	0.2093	0.0706	0.2163
	200	0.0672	0.1781	0.0662	0.1813	0.0685	0.1867
SE	25	0.1298	1.0254	0.0169	0.6801	0.3637	0.7047
	50	0.0934	0.5478	0.0110	0.4933	0.1190	0.2945
	100	0.1629	0.3002	0.1485	0.2666	0.1052	0.1858
	150	0.1206	0.2731	0.0512	0.2351	0.0928	0.1529
	200	0.1736	0.2623	0.0736	0.1853	0.1457	0.1497
JSE	25	0.3167	4.7204	0.4594	6.0041	0.4751	5.5051
	50	0.1586	3.1748	0.1988	5.7254	0.3179	5.3947
	100	0.1965	2.1097	0.3427	5.0111	0.1710	5.3683
	150	0.1029	2.0062	0.2095	4.7791	0.2699	5.3016
	200	0.2396	1.7858	0.2952	4.7019	0.1746	5.1026
RSE	25	0.0100	0.5285	0.0157	0.5742	0.0415	0.6514
	50	0.0161	0.2630	0.0246	0.3027	0.0195	0.3105
	100	0.0236	0.1779	0.0211	0.1759	0.0182	0.1691
	150	0.0214	0.1315	0.0182	0.1328	0.0213	0.1359
	200	0.0190	0.1099	0.0214	0.1128	0.0204	0.1124

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A20. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 10% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.7944	5.2375	0.5017	6.1350	0.6275	6.4083
	50	0.5521	4.8550	0.7120	5.9626	0.4389	6.2877
	100	0.8544	4.1828	0.4655	5.7585	0.6999	6.0153
	150	0.7160	3.0651	0.5376	5.3560	0.5693	5.2649
	200	0.5921	2.9390	0.5933	5.2213	0.6371	5.0758
HME	25	0.0695	0.5643	0.0608	0.5857	0.0679	0.8973
	50	0.0690	0.3065	0.0660	0.2964	0.0626	0.3229
	100	0.0623	0.2039	0.0598	0.2038	0.0604	0.2116
	150	0.0600	0.1724	0.0624	0.1757	0.0629	0.1757
	200	0.0614	0.1599	0.0619	0.1614	0.0601	0.1588
MME	25	0.0562	0.5725	0.0457	0.5904	0.0489	1.2000
	50	0.0562	0.2944	0.0548	0.2909	0.0484	0.3485
	100	0.0468	0.1900	0.0464	0.1906	0.0450	0.1981
	150	0.0458	0.1560	0.0484	0.1639	0.0481	0.1609
	200	0.0463	0.1433	0.0448	0.1420	0.0452	0.1447
LMSE	25	0.0625	1.1280	0.0498	1.1404	0.0931	3.3462
	50	0.0519	0.6491	0.0658	0.6595	0.0648	1.3806
	100	0.0548	0.4080	0.0576	0.4131	0.0598	0.5696
	150	0.0529	0.3264	0.0605	0.3239	0.0503	0.3914
	200	0.0566	0.2774	0.0519	0.2713	0.0580	0.3142
LTSE	25	0.0973	0.6964	0.0647	0.6953	0.0736	1.6717
	50	0.0780	0.3913	0.0783	0.3967	0.0724	0.6050
	100	0.0750	0.2562	0.0680	0.2529	0.0700	0.3075
	150	0.0673	0.2046	0.0731	0.2093	0.0706	0.2163
	200	0.0672	0.1781	0.0662	0.1813	0.0685	0.1867
SE	25	0.1298	1.0254	0.0169	0.6801	0.3637	0.7047
	50	0.0934	0.5478	0.0110	0.4933	0.1190	0.2945
	100	0.1629	0.3002	0.1485	0.2666	0.1052	0.1858
	150	0.1206	0.2731	0.0512	0.2351	0.0928	0.1529
	200	0.1736	0.2623	0.0736	0.1853	0.1457	0.1497
JSE	25	0.3167	4.7204	0.4594	6.0041	0.4751	5.5051
	50	0.1586	3.1748	0.1988	5.7254	0.3179	5.3947
	100	0.1965	2.1097	0.3427	5.0111	0.1710	5.3683
	150	0.1029	2.0062	0.2095	4.7791	0.2699	5.3016
	200	0.2396	1.7858	0.2952	4.7019	0.1746	5.1026
RSE	25	0.0100	0.5285	0.0157	0.5742	0.0415	0.6514
	50	0.0161	0.2630	0.0246	0.3027	0.0195	0.3105
	100	0.0236	0.1779	0.0211	0.1759	0.0182	0.1691
	150	0.0214	0.1315	0.0182	0.1328	0.0213	0.1359
	200	0.0190	0.1099	0.0214	0.1128	0.0204	0.1124

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A21. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 25% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.7197	6.1436	0.4972	6.6404	0.6248	6.5710
	50	0.6168	6.0352	0.6490	6.6169	0.5822	6.1232
	100	0.6384	5.3854	0.6268	6.1234	0.5426	6.0692
	150	0.6185	5.0035	0.5820	5.4785	0.6485	5.2365
	200	0.7964	4.5073	0.5438	5.0980	0.5591	5.0336
HME	25	0.0624	0.3609	0.0500	0.3594	0.0579	0.3638
	50	0.0679	0.2331	0.0592	0.2360	0.0594	0.2316
	100	0.0604	0.1783	0.0564	0.1780	0.0620	0.1811
	150	0.0606	0.1604	0.0569	0.1595	0.0575	0.1577
	200	0.0629	0.1523	0.0601	0.1527	0.0627	0.1536
MME	25	0.0510	0.3599	0.0474	0.3606	0.0449	0.3565
	50	0.0530	0.2201	0.0519	0.2177	0.0513	0.2196
	100	0.0461	0.1625	0.0468	0.1615	0.0462	0.1618
	150	0.0477	0.1434	0.0471	0.1435	0.0449	0.1427
	200	0.0460	0.1335	0.0469	0.1348	0.0452	0.1334
LMSE	25	0.0571	0.8234	0.0557	0.8147	0.0521	0.8229
	50	0.0541	0.4482	0.0580	0.4417	0.0566	0.4412
	100	0.0537	0.2613	0.0540	0.2610	0.0550	0.2635
	150	0.0578	0.2292	0.0558	0.2290	0.0556	0.2292
	200	0.0543	0.2115	0.0564	0.2134	0.0559	0.2125
LTSE	25	0.0744	0.4775	0.0732	0.4801	0.0670	0.4804
	50	0.0741	0.3025	0.0734	0.3042	0.0713	0.3037
	100	0.0702	0.2001	0.0678	0.1998	0.0684	0.2009
	150	0.0665	0.1720	0.0687	0.1716	0.0697	0.1719
	200	0.0709	0.1635	0.0701	0.1639	0.0699	0.1644
SE	25	0.1102	0.5024	0.1217	0.5035	0.0173	0.4901
	50	0.2590	0.4799	0.1471	0.3330	0.1527	0.4601
	100	0.1934	0.2489	0.1479	0.2863	0.1231	0.2918
	150	0.1387	0.2201	0.1286	0.2488	0.1150	0.2446
	200	0.1072	0.1870	0.1130	0.1855	0.1092	0.1880
JSE	25	0.2322	7.0444	0.2406	8.1936	0.2621	8.1735
	50	0.2676	7.0099	0.1940	8.0740	0.2975	7.5860
	100	0.2418	6.5458	0.2174	7.5776	0.2899	7.4900
	150	0.2262	6.1071	0.3186	6.7061	0.3452	7.0690
	200	0.2661	5.4648	0.2850	5.7614	0.3689	6.2156
RSE	25	0.0103	0.3262	0.0073	0.3252	0.0091	0.3206
	50	0.0276	0.2069	0.0077	0.2121	0.0054	0.2102
	100	0.0214	0.1277	0.0082	0.1278	0.0062	0.1306
	150	0.0215	0.1008	0.0079	0.1000	0.0087	0.1005
	200	0.0188	0.0869	0.0079	0.0880	0.0073	0.0878

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A21. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 25% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.7197	6.1436	0.4972	6.6404	0.6248	6.5710
	50	0.6168	6.0352	0.6490	6.6169	0.5822	6.1232
	100	0.6384	5.3854	0.6268	6.1234	0.5426	6.0692
	150	0.6185	5.0035	0.5820	5.4785	0.6485	5.2365
	200	0.7964	4.5073	0.5438	5.0980	0.5591	5.0336
HME	25	0.0624	0.3609	0.0500	0.3594	0.0579	0.3638
	50	0.0679	0.2331	0.0592	0.2360	0.0594	0.2316
	100	0.0604	0.1783	0.0564	0.1780	0.0620	0.1811
	150	0.0606	0.1604	0.0569	0.1595	0.0575	0.1577
	200	0.0629	0.1523	0.0601	0.1527	0.0627	0.1536
MME	25	0.0510	0.3599	0.0474	0.3606	0.0449	0.3565
	50	0.0530	0.2201	0.0519	0.2177	0.0513	0.2196
	100	0.0461	0.1625	0.0468	0.1615	0.0462	0.1618
	150	0.0477	0.1434	0.0471	0.1435	0.0449	0.1427
	200	0.0460	0.1335	0.0469	0.1348	0.0452	0.1334
LMSE	25	0.0571	0.8234	0.0557	0.8147	0.0521	0.8229
	50	0.0541	0.4482	0.0580	0.4417	0.0566	0.4412
	100	0.0537	0.2613	0.0540	0.2610	0.0550	0.2635
	150	0.0578	0.2292	0.0558	0.2290	0.0556	0.2292
	200	0.0543	0.2115	0.0564	0.2134	0.0559	0.2125
LTSE	25	0.0744	0.4775	0.0732	0.4801	0.0670	0.4804
	50	0.0741	0.3025	0.0734	0.3042	0.0713	0.3037
	100	0.0702	0.2001	0.0678	0.1998	0.0684	0.2009
	150	0.0665	0.1720	0.0687	0.1716	0.0697	0.1719
	200	0.0709	0.1635	0.0701	0.1639	0.0699	0.1644
SE	25	0.1102	0.5024	0.1217	0.5035	0.0173	0.4901
	50	0.2590	0.4799	0.1471	0.3330	0.1527	0.4601
	100	0.1934	0.2489	0.1479	0.2863	0.1231	0.2918
	150	0.1387	0.2201	0.1286	0.2488	0.1150	0.2446
	200	0.1072	0.1870	0.1130	0.1855	0.1092	0.1880
JSE	25	0.2322	7.0444	0.2406	8.1936	0.2621	8.1735
	50	0.2676	7.0099	0.1940	8.0740	0.2975	7.5860
	100	0.2418	6.5458	0.2174	7.5776	0.2899	7.4900
	150	0.2262	6.1071	0.3186	6.7061	0.3452	7.0690
	200	0.2661	5.4648	0.2850	5.7614	0.3689	6.2156
RSE	25	0.0103	0.3262	0.0073	0.3252	0.0091	0.3206
	50	0.0276	0.2069	0.0077	0.2121	0.0054	0.2102
	100	0.0214	0.1277	0.0082	0.1278	0.0062	0.1306
	150	0.0215	0.1008	0.0079	0.1000	0.0087	0.1005
	200	0.0188	0.0869	0.0079	0.0880	0.0073	0.0878

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A22. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 50% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.5657	7.9980	0.4434	8.9370	0.6042	8.0520
	50	0.7596	7.0075	0.5667	8.2005	0.5769	7.9962
	100	0.8088	6.2413	0.5829	7.3920	0.5820	7.7281
	150	0.6682	5.4902	0.7837	6.4956	0.6723	6.4396
	200	0.7307	5.0436	0.8332	5.3495	0.5592	5.6593
HME	25	0.0657	0.2953	0.0582	0.2969	0.0578	0.2917
	50	0.0612	0.2139	0.0654	0.2124	0.0623	0.2131
	100	0.0651	0.1721	0.0580	0.1679	0.0612	0.1758
	150	0.0612	0.1567	0.0627	0.1582	0.0614	0.1559
	200	0.0645	0.1484	0.0615	0.1483	0.0593	0.1483
MME	25	0.0479	0.2803	0.0449	0.2840	0.0449	0.2810
	50	0.0480	0.1995	0.0512	0.1984	0.0479	0.1999
	100	0.0499	0.1552	0.0454	0.1518	0.0434	0.1575
	150	0.0496	0.1415	0.0455	0.1373	0.0473	0.1378
	200	0.0464	0.1283	0.0461	0.1263	0.0451	0.1296
LMSE	25	0.0594	0.5024	0.0572	0.5109	0.0464	0.4924
	50	0.0536	0.3142	0.0523	0.3122	0.0626	0.3105
	100	0.0548	0.2374	0.0546	0.2300	0.0526	0.2314
	150	0.0556	0.2082	0.0595	0.2112	0.0561	0.2105
	200	0.0541	0.1899	0.0557	0.1919	0.0562	0.1953
LTSE	25	0.0713	0.3348	0.0669	0.3417	0.0678	0.3342
	50	0.0733	0.2375	0.0690	0.2358	0.0676	0.2295
	100	0.0693	0.1848	0.0699	0.1838	0.0657	0.1865
	150	0.0706	0.1679	0.0708	0.1707	0.0684	0.1704
	200	0.0701	0.1584	0.0699	0.1608	0.0704	0.1616
SE	25	0.0487	0.3743	0.1390	0.3623	0.0858	0.3725
	50	0.1429	0.3441	0.1533	0.2081	0.0967	0.2990
	100	0.0755	0.2189	0.1182	0.2116	0.1140	0.2466
	150	0.1122	0.1808	0.1004	0.2062	0.1067	0.2130
	200	0.1100	0.1816	0.1044	0.1792	0.0577	0.1935
JSE	25	0.2102	9.2222	0.2572	9.9901	0.1209	9.5601
	50	0.0821	8.6237	0.2206	9.6870	0.1603	9.3979
	100	0.3418	7.7882	0.2696	8.9530	0.2304	9.0038
	150	0.3379	6.8491	0.1748	7.9881	0.3194	8.6317
	200	0.0998	5.1003	0.3074	6.8933	0.4819	7.2279
RSE	25	0.0075	0.2611	0.0061	0.2718	0.0082	0.2668
	50	0.0039	0.1622	0.0110	0.1645	0.0030	0.1593
	100	0.0051	0.1044	0.0039	0.1027	0.0057	0.1072
	150	0.0045	0.0854	0.0072	0.0843	0.0054	0.0828
	200	0.0062	0.0730	0.0042	0.0732	0.0051	0.0735

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).

Table A22. Estimated Bias and RMSE Values for ${\epsilon }_i\sim t_2$ with 50% Outliers in the x Direction and Multicollinearity.

Method	$\mathit{n}$	$\mathit{\rho }=0.10$		$\mathit{\rho }=0.50$		$\mathit{\rho }=0.98$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
OLSE	25	0.5657	7.9980	0.4434	8.9370	0.6042	8.0520
	50	0.7596	7.0075	0.5667	8.2005	0.5769	7.9962
	100	0.8088	6.2413	0.5829	7.3920	0.5820	7.7281
	150	0.6682	5.4902	0.7837	6.4956	0.6723	6.4396
	200	0.7307	5.0436	0.8332	5.3495	0.5592	5.6593
HME	25	0.0657	0.2953	0.0582	0.2969	0.0578	0.2917
	50	0.0612	0.2139	0.0654	0.2124	0.0623	0.2131
	100	0.0651	0.1721	0.0580	0.1679	0.0612	0.1758
	150	0.0612	0.1567	0.0627	0.1582	0.0614	0.1559
	200	0.0645	0.1484	0.0615	0.1483	0.0593	0.1483
MME	25	0.0479	0.2803	0.0449	0.2840	0.0449	0.2810
	50	0.0480	0.1995	0.0512	0.1984	0.0479	0.1999
	100	0.0499	0.1552	0.0454	0.1518	0.0434	0.1575
	150	0.0496	0.1415	0.0455	0.1373	0.0473	0.1378
	200	0.0464	0.1283	0.0461	0.1263	0.0451	0.1296
LMSE	25	0.0594	0.5024	0.0572	0.5109	0.0464	0.4924
	50	0.0536	0.3142	0.0523	0.3122	0.0626	0.3105
	100	0.0548	0.2374	0.0546	0.2300	0.0526	0.2314
	150	0.0556	0.2082	0.0595	0.2112	0.0561	0.2105
	200	0.0541	0.1899	0.0557	0.1919	0.0562	0.1953
LTSE	25	0.0713	0.3348	0.0669	0.3417	0.0678	0.3342
	50	0.0733	0.2375	0.0690	0.2358	0.0676	0.2295
	100	0.0693	0.1848	0.0699	0.1838	0.0657	0.1865
	150	0.0706	0.1679	0.0708	0.1707	0.0684	0.1704
	200	0.0701	0.1584	0.0699	0.1608	0.0704	0.1616
SE	25	0.0487	0.3743	0.1390	0.3623	0.0858	0.3725
	50	0.1429	0.3441	0.1533	0.2081	0.0967	0.2990
	100	0.0755	0.2189	0.1182	0.2116	0.1140	0.2466
	150	0.1122	0.1808	0.1004	0.2062	0.1067	0.2130
	200	0.1100	0.1816	0.1044	0.1792	0.0577	0.1935
JSE	25	0.2102	9.2222	0.2572	9.9901	0.1209	9.5601
	50	0.0821	8.6237	0.2206	9.6870	0.1603	9.3979
	100	0.3418	7.7882	0.2696	8.9530	0.2304	9.0038
	150	0.3379	6.8491	0.1748	7.9881	0.3194	8.6317
	200	0.0998	5.1003	0.3074	6.8933	0.4819	7.2279
RSE	25	0.0075	0.2611	0.0061	0.2718	0.0082	0.2668
	50	0.0039	0.1622	0.0110	0.1645	0.0030	0.1593
	100	0.0051	0.1044	0.0039	0.1027	0.0057	0.1072
	150	0.0045	0.0854	0.0072	0.0843	0.0054	0.0828
	200	0.0062	0.0730	0.0042	0.0732	0.0051	0.0735

Note: OLSE (Ordinary Least Squares Estimator), HME (Huber Maximum Likelihood Estimator), MME (Modified Maximum Likelihood Estimator), LMSE (Least Median of Squares Estimator), and LTSE (Least Trimmed Squares Estimator). SE (S-Estimator), JSE (James–Stein Estimator), and RSE (Robust Stein Estimator).