Bias-Corrected and Variance-Corrected MLE for The New Median Based Unit Weibull Distribution (MBUW)

1. Section 1

Bias-Corrected MLE

Let $\mathsf{\Theta }$ be a p-dimensional unknown parameter vector and $l=l\left(\mathsf{\Theta }|y\right)$ be the log-likelihood function for a sample of n observations. Assume this log-likelihood is regular with respect to all derivatives up to and including those of third order. The joint cumulants of the log-likelihood derivatives are defined as follows in equations (4-6)

$$k_{ij}=E\left({\displaystyle \frac{{\partial }^{2}l}{\partial {\mathsf{\Theta }}_i\partial {\mathsf{\Theta }}_j}}\right) ;i,j=\mathrm{1,2},\dots .,p$$

(4)

$$k_{ijl}=E\left({\displaystyle \frac{{\partial }^{3}l}{\partial {\mathsf{\Theta }}_i\partial {\mathsf{\Theta }}_j\partial {\mathsf{\Theta }}_l}}\right); i,j=\mathrm{1,2},\dots .,p$$

(5)

$$k_{ij,l}=E\left({\displaystyle \frac{{\partial }^{2}l}{\partial {\mathsf{\Theta }}_i\partial {\mathsf{\Theta }}_j}} {\displaystyle \frac{{\partial }^{3}l}{\partial {\mathsf{\Theta }}_l}}\right) ; i,j=\mathrm{1,2},\dots .,p$$

(6)

The derivative of these cumulants are defined as in equation (7)

$$k_{ij}^{\left(l\right)}=E\left({\displaystyle \frac{\partial k_{ij}}{\partial {\mathsf{\Theta }}_l}}\right);i,j,l=\mathrm{1,2},\dots ..,p$$

(7)

The expressions in eqautions (4-7) are assumed to be $O\left(n\right)$ .

(Cox & Snell, 1968) revealed that when the sample data are independent (but not necessarily identically distributed) the bias of the s^th element of the MLE of ${\widehat{\mathsf{\Theta }}}_s$, is calculated as

$$Bias\left({\widehat{\mathsf{\Theta }}}_s\right)=\sum _{i=1}^p\sum _{j=1}^p\sum _{l=1}^p{k}^{si}{k}^{jl}\left[0.5\ast k_{ijl}+k_{ij,l}\right]+O\left(n^{-2}\right)$$

(8)

where : $s=1,\dots .,p$ and $k^{ij}$is the $(i,j)th$ element of the inverse of the expected information matrix $K=-k_{ij}$ .

(Cordeiro & Klein, 1994) consequently established that equation (8) yet maintains if the data are non-independent and that it can be disclosed as

$$Bias\left({\widehat{\mathsf{\Theta }}}_s\right)=\sum _{i=1}^p{k}^{si}\sum _{j=1}^p\sum _{l=1}^p\left[k_{ij}^{\left(l\right)}-0.5\ast k_{ijl}\right]k^{jl}+O\left(n^{-2}\right)$$

(9)

The bias equation in (9) is largely simpler to calculate than in (8), because it does not include terms of the form given in (6).

Defining the following terms in equations (10-11):

$$a_{ij}^{\left(l\right)}=k_{ij}^{\left(l\right)}-0.5k_{ijl} ; i,j,l=\mathrm{1,2},\dots ,p$$

(10)

$$A^{\left(l\right)}=\left\{a_{ij}^{\left(l\right)}\right\} ; i,j,l=\mathrm{1,2},\dots ,p$$

(11)

and collecting terms up into matrices $A=\left[ A^{\left(1\right)}|\dots | A^{\left(p\right)} \right]$

The $O\left(n^{-2}\right)$ bias of the MLE of $\widehat{\mathsf{\Theta }}$ in (9) can be rephrased in the convenient form equation (12):

$$Bias\left({\widehat{\mathsf{\Theta }}}_s\right)={\widehat{K}}^{-1}\widehat{A} vec \left({\widehat{K}}^{-1}\right)$$

(12)

where $\widehat{K}=K{|}_{\mathsf{\Theta }=\widehat{\mathsf{\Theta }}}$ and $\widehat{A}=A{|}_{\mathsf{\Theta }=\widehat{\mathsf{\Theta }}}$, the value of ${\widehat{\mathsf{\Theta }}}_s$ is obtained by solving the roots of log-likelihood equations using the numerical methods. $Vec\left(.\right)$ means vectorization operator, which stacks the columns of the matrix in question one above the other, forming one extended column vector. Hence the bias adjusted-MLE is defined in equation (13) as

$$\tilde{\mathsf{\Theta }}=\widehat{\mathsf{\Theta }}-{\widehat{K}}^{-1}\widehat{A} vec \left({\widehat{K}}^{-1}\right)$$

(13)

One of the benefits of this method is that these expressions can be calculated when solving for the root of the log-likelihood equations do not disclose an analytic closed-form solution. In such circumstances, the bias-corrected MLE can be easily obtained by means of standard numerical methods, and $\tilde{\mathsf{\Theta }}$ is unbiased $O\left(n^{-2}\right)$

2. Section 2

Bias Reduction and the MBUW

The PDF of MBUW satisfies the regularity conditions. The first order partial derivatives of log-likelihood with respect to alpha and beta parameters are defined in equation (14-15) as follows:

$${\displaystyle \frac{\partial l}{\partial \alpha }}={\displaystyle \frac{\beta }{\alpha }}+\sum _{i=1}^n{\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta \left(\ln y\right){\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}-2\beta {\alpha }^{\left(-\beta -1\right)} \sum _{i=1}^n\ln y$$

(14)

$${\displaystyle \frac{\partial l}{\partial \beta }}=-n\ln \alpha +\sum _{i=1}^n{\displaystyle \frac{y^{{\alpha }^{-\beta }}{\alpha }^{-\beta }\left(\ln y\right)\left(\ln \alpha \right)}{1-y^{{\alpha }^{-\beta }}}}-2{\alpha }^{-\beta } \left(\ln \alpha \right)\sum _{i=1}^n\ln y$$

(15)

Equations (16-24) are the higher order derivatives (for one observation):

$$\begin{gathered} {\displaystyle \frac{{\partial }^{2}l}{\partial {\alpha }^2}}={\displaystyle \frac{\beta }{{\alpha }^2}}-\left( {\displaystyle \frac{y^{{\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}} \right)-{\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta \left(\beta +1\right)\left(\ln y\right){\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}} \\ -\left( {\displaystyle \frac{y^{{2\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}} \right)+2\beta \left(\beta +1\right){\alpha }^{\left(-\beta -2\right)}\left(\ln y\right) \end{gathered}$$

(16)

$$\begin{gathered} {\displaystyle \frac{{\partial }^{2}l}{\partial {\beta }^2}}=\left( {\displaystyle \frac{{-y}^{{\alpha }^{-\beta }}{\left(\ln \alpha \right)}^2{\left(\ln y\right)}^2{\alpha }^{\left(-2\beta \right)}}{1-y^{{\alpha }^{-\beta }}}} \right)-\left({\displaystyle \frac{{ y}^{{\alpha }^{-\beta }}{\left(\ln \alpha \right)}^2\left(\ln y\right){\alpha }^{\left(-\beta \right)}}{1-y^{{\alpha }^{-\beta }}}}\right) \\ -\left( {\displaystyle \frac{y^{{2\alpha }^{-\beta }}{\left(\ln \alpha \right)}^2{\left(\ln y\right)}^2{\alpha }^{\left(-2\beta \right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}} \right)+2{\alpha }^{\left(-\beta \right)}\left(\ln y\right){\left(\ln \alpha \right)}^2 \end{gathered}$$

(17)

$$\begin{gathered} {\displaystyle \frac{{\partial }^{2}l}{\partial \alpha \partial \beta }}={\displaystyle \frac{-1}{\alpha }}-\left( {\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta \left(\ln \alpha \right){\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}} \right)+{\displaystyle \frac{y^{{\alpha }^{-\beta }}\left(\ln y\right){\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}} \\ -\left( {\displaystyle \frac{{ y}^{{\alpha }^{-\beta }}\beta \left(\ln \alpha \right)\left(\ln y\right){\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}} \right)-\left( {\displaystyle \frac{y^{{2\alpha }^{-\beta }}\beta \left(\ln \alpha \right){\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}} \right) \\ -2{\alpha }^{\left(-\beta -1\right)} \left(\ln y\right)+2 \beta {\alpha }^{\left(-\beta -1\right)} \left(\ln \alpha \right)\left(\ln y\right) \end{gathered}$$

(18)

$${\displaystyle \frac{{\partial }^{3}l}{\partial {\alpha }^2\partial \beta }}={\displaystyle \frac{1}{{\alpha }^2}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^3 \left(\ln \alpha \right) {\alpha }^{\left(-3\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{{2 y}^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{{2y}^{{\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{2\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^3 \left(\ln \alpha \right) {\alpha }^{\left(-3\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$-{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left(2\beta +1\right) \left(\ln y\right) {\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left({\beta }^2+\beta \right) \left(\ln \alpha \right) {\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left({\beta }^2+\beta \right) \left(\ln \alpha \right) \left(\ln y\right){\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{2\alpha }^{-\beta }}\left({\beta }^2+\beta \right){\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{{2 y}^{{2\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^3 \left(\ln \alpha \right) {\alpha }^{\left(-3\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}-{\displaystyle \frac{{2y}^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{{2 y}^{{2\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{2y}^{{3\alpha }^{-\beta }}{\beta }^2{ \left(\ln \alpha \right) \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}}$$

$$+4\beta {\alpha }^{\left(-\beta -2\right)} \left(\ln y\right)+2{\alpha }^{\left(-\beta -2\right)} \left(\ln y\right)-2 {\beta }^2{\alpha }^{\left(-\beta -2\right)} \left(\ln y\right)\left(\ln \alpha \right)$$

$$-2 \beta {\alpha }^{\left(-\beta -2\right)} \left(\ln y\right)\left(\ln \alpha \right)$$

(19)

$${\displaystyle \frac{{\partial }^{2}l}{\partial {\beta }^2\partial \alpha }}={\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^3 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-3\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{2 y}^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$-{\displaystyle \frac{{2y}^{{\alpha }^{-\beta }}{\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^3 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-3\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left(\beta \right) \left(\ln y\right) {\left(\ln \alpha \right)}^2{\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$-{\displaystyle \frac{{2y}^{{\alpha }^{-\beta }} \left(\ln \alpha \right) \left(\ln y\right){\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{y^{{2\alpha }^{-\beta }}\left(\beta \right){\left(\ln y\right)}^2 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{{2 y}^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^3 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-3\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{2y}^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 {\left(\ln \alpha \right)}^2{\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$-{\displaystyle \frac{{2 y}^{{2\alpha }^{-\beta }}{\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{2y}^{{3\alpha }^{-\beta }}\beta { {\left(\ln \alpha \right)}^2 \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}}$$

$$-2 \beta {\alpha }^{\left(-\beta -1\right)} \left(\ln y\right){\left(\ln \alpha \right)}^2+4 {\alpha }^{\left(-\beta -1\right)} \left(\ln y\right) \left(\ln \alpha \right)$$

(20)

$${\displaystyle \frac{{\partial }^{3}l}{\partial {\alpha }^3}}={\displaystyle \frac{-2\beta }{{\alpha }^3}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }}{\beta }^3{\left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -3\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{3 y}^{{\alpha }^{-\beta }}{\beta }^{2 }\left(\beta +1\right){\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta -3\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{{3{\beta }^{3}y}^{{2\alpha }^{-\beta }}{\left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -3\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta \left(\beta +1\right)\left(\beta +2\right) \left(\ln y\right) {\alpha }^{\left(-\beta -3\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{{3{\beta }^2 \left(\beta +1\right) y}^{{2\alpha }^{-\beta }}{\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta -3\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{2y}^{{3\alpha }^{-\beta }}{\beta }^3{ \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -3\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}}$$

$$-2 \beta \left(\beta +1\right)\left(\beta +2\right) {\alpha }^{\left(-\beta -3\right)} \left(\ln y\right)$$

(21)

$${\displaystyle \frac{{\partial }^{3}l}{\partial {\beta }^3}}={\displaystyle \frac{y^{{\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3\left(\ln y\right) {\alpha }^{\left(-\beta \right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{3y}^{{\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3{\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta \right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3{\left(\ln y\right)}^3 {\alpha }^{\left(-3\beta \right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{3y}^{{2\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3{\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta \right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{{3y}^{{2\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3{\left(\ln y\right)}^3 {\alpha }^{\left(-3\beta \right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{2y}^{{3\alpha }^{-\beta }}{\left(\ln \alpha \right)}^3{ \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta \right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}}$$

$$-2 {\alpha }^{\left(-\beta \right)} \left(\ln y\right){\left(\ln \alpha \right)}^3$$

(22)

$${\displaystyle \frac{{\partial }^{3}l}{\partial \beta \partial \alpha \partial \alpha }}={\displaystyle \frac{1}{{\alpha }^2}}-2 \beta {\left(\beta +1\right)\alpha }^{\left(-\beta -2\right)} \left(\ln y\right)\left(\ln \alpha \right)+2{\left(2\beta +1\right)\alpha }^{\left(-\beta -2\right)}\left(\ln y\right)$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }}\beta {\left(2\beta +1\right)\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{{ y}^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^3 \left(\ln \alpha \right) {\alpha }^{\left(-3\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{3y}^{{2\alpha }^{-\beta }}{\beta }^2{\left(\ln y\right)}^3 \left(\ln \alpha \right) {\alpha }^{\left(-3\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$-{\displaystyle \frac{y^{{\alpha }^{-\beta }} \beta {\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left(\beta +1\right) \left(\ln y\right) {\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$-{\displaystyle \frac{{3y}^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{y^{{\alpha }^{-\beta }} \beta \left(1+\beta \right) \left(\ln \alpha \right) \left(\ln y\right){\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }} {\beta }^2 \left(\ln \alpha \right) {\left(\ln y\right)}^2{\alpha }^{\left(-2\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{y^{{\alpha }^{-\beta }} \beta \left(\ln y\right){\alpha }^{\left(-\beta -2\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{{ y}^{{2\alpha }^{-\beta }}{\beta }^2{ \left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}+{\displaystyle \frac{{\beta y}^{{2\alpha }^{-\beta }}\left(2\beta +1\right){\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{{2y}^{{3\alpha }^{-\beta }}{\beta }^2{ \left(\ln \alpha \right) \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -2\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}}$$

(23)

$${\displaystyle \frac{{\partial }^{3}l}{\partial \alpha \partial \beta \partial \beta }}=4{ \alpha }^{\left(-\beta -1\right)}\left(\ln y\right)\left(\ln \alpha \right)-2\beta { \alpha }^{\left(-\beta -1\right)}\left(\ln y\right){\left(\ln \alpha \right)}^2$$

$$+{\displaystyle \frac{{ \beta y}^{{\alpha }^{-\beta }}{ {\left(\ln \alpha \right)}^2 \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{{2y}^{{\alpha }^{-\beta }}{\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}$$

$$+{\displaystyle \frac{{ 3y}^{{\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2{ {\left(\ln \alpha \right)}^2 \alpha }^{\left(-2\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{3y}^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^3 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-3\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$-{\displaystyle \frac{2y^{{\alpha }^{-\beta }} \left(\ln \alpha \right) \left(\ln y\right) {\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}-{\displaystyle \frac{2y^{{2\alpha }^{-\beta }}\beta {\left(\ln y\right)}^2 \left(\ln \alpha \right) {\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+{\displaystyle \frac{y^{{\alpha }^{-\beta }} \beta \left(\ln y\right) {\left(\ln \alpha \right)}^2{\alpha }^{\left(-\beta -1\right)}}{1-y^{{\alpha }^{-\beta }}}}+{\displaystyle \frac{{ 3y}^{{2\alpha }^{-\beta }}\beta { \left(\ln y\right)}^2 {\left(\ln \alpha \right)}^2 {\alpha }^{\left(-2\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^2}}$$

$$+\frac{{2 \beta y}^{{3\alpha }^{-\beta }}{ {\left(\ln \alpha \right)}^2 \left(\ln y\right)}^3 {\alpha }^{\left(-3\beta -1\right)}}{{\left(1-y^{{\alpha }^{-\beta }}\right)}^3}$$

(24)

Taking expectation of the above derivative is carried out using Monte Carlo integration; see equations (25-31). The following integral can be calculated using Monte Carlo integration and it is a fixed number. The author underwent trapezoid method for integration. Substituting the estimated parameters alpha and beta for each data set, the following integrals can be calculated and are fixed value for each dataset.

$$E\left({\displaystyle \frac{y^{{\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{1-y^{{\alpha }^{-\beta }}}}\right)={\int }_0^1{\displaystyle \frac{y^{{\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{1-y^{{\alpha }^{-\beta }}}}f\left(y\right)dy=f_1$$

(25)

$$E\left({\displaystyle \frac{y^{{\alpha }^{-\beta }} {\left(\ln y\right)}^2 }{1-y^{{\alpha }^{-\beta }}}}\right)={\int }_0^1{\displaystyle \frac{y^{{\alpha }^{-\beta }} {\left(\ln y\right)}^2 }{1-y^{{\alpha }^{-\beta }}}}f\left(y\right)dy=f_2$$

(26)

$$E\left({\displaystyle \frac{y^{{\alpha }^{-\beta }} \ln y }{1-y^{{\alpha }^{-\beta }}}}\right)={\int }_0^1{\displaystyle \frac{y^{{\alpha }^{-\beta }} \left(\ln y\right) }{1-y^{{\alpha }^{-\beta }}}}f\left(y\right)dy=f_3$$

(27)

$$\left({\displaystyle \frac{y^{{2\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^2}}\right)={\int }_0^1{\displaystyle \frac{y^{{2\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^2}}f\left(y\right)dy=f_4$$

(28)

$$E\left({\displaystyle \frac{y^{{2\alpha }^{-\beta }} {\left(\ln y\right)}^2 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^2}}\right)={\int }_0^1{\displaystyle \frac{y^{{2\alpha }^{-\beta }} {\left(\ln y\right)}^2 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^2}}f\left(y\right)dy=f_5$$

(29)

$$E\left({\displaystyle \frac{y^{{3\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^3}}\right)={\int }_0^1{\displaystyle \frac{y^{{3\alpha }^{-\beta }} {\left(\ln y\right)}^3 }{{\left[1-y^{{\alpha }^{-\beta }}\right]}^3}}f\left(y\right)dy=f_6$$

(30)

$$E\left(\ln y\right)={\int }_0^1\left(\ln y\right)f\left(y\right)dy=f_7$$

(31)

Define the following quantities:

$$k_{11 }=E \left( {\displaystyle \frac{{\partial }^{2}l}{\partial {\alpha }^2}}\right), k_{22 }=E \left({\displaystyle \frac{{\partial }^{2}l}{\partial {\beta }^2}}\right) , k_{12 }=E \left({\displaystyle \frac{{\partial }^{2}l}{\partial \alpha \partial \beta }}\right)=k_{21}=E \left({\displaystyle \frac{{\partial }^{2}l}{\partial \beta \partial \alpha }}\right)$$

$$k_{111 }=E \left( {\displaystyle \frac{{\partial }^{3}l}{\partial {\alpha }^3}}\right) , k_{222 }=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial {\beta }^3}}\right), k_{121}=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial \alpha \partial \beta \partial \alpha }}\right)=k_{211}=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial \beta \partial \alpha \partial \alpha }}\right)$$

$$k_{122}=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial \alpha \partial \beta \partial \beta }}\right)=k_{212}=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial \beta \partial \alpha \partial \beta }}\right)$$

$$k_{112 }=E \left( {\displaystyle \frac{{\partial }^{3}l}{\partial {\alpha }^2\partial \beta }}\right) , k_{221 }=E \left({\displaystyle \frac{{\partial }^{3}l}{\partial {\beta }^2\partial \alpha }}\right)$$

$$k_{11}^{\left(1\right)}={\displaystyle \frac{\partial k_{11}}{\partial \alpha }} , k_{12}^{\left(1\right)}={\displaystyle \frac{\partial k_{12}}{\partial \alpha }}=k_{21}^{\left(1\right)}={\displaystyle \frac{\partial k_{21}}{\partial \alpha }} , k_{22}^{\left(1\right)}={\displaystyle \frac{\partial k_{22}}{\partial \alpha }}$$

$$k_{11}^{\left(2\right)}={\displaystyle \frac{\partial k_{11}}{\partial \beta }} , k_{12}^{\left(2\right)}={\displaystyle \frac{\partial k_{12}}{\partial \beta }}=k_{21}^{\left(2\right)}={\displaystyle \frac{\partial k_{21}}{\partial \beta }} , k_{22}^{\left(2\right)}={\displaystyle \frac{\partial k_{22} }{\partial \beta }}$$

$$a_{11}^{\left(1\right)}=k_{11}^{\left(1\right)}-0.5 k_{111}$$

$$a_{12}^{\left(1\right)}=k_{12}^{\left(1\right)}-0.5 k_{121}$$

$$a_{22}^{\left(1\right)}=k_{22}^{\left(1\right)}-0.5 k_{221}$$

$$a_{11}^{\left(2\right)}=k_{11}^{\left(2\right)}-0.5 k_{112}$$

$$a_{12}^{\left(2\right)}=k_{12}^{\left(2\right)}-0.5 k_{122}$$

$$a_{22}^{\left(2\right)}=k_{22}^{\left(2\right)}-0.5 k_{222}$$

The information matrix is $K=\left\{-k_{ij}\right\}=-n\times \left[\begin{array}{cc}{k}_{11}& k_{12}\\ k_{21}& k_{22}\end{array}\right],$where n is the number of observations.

Defining $A_{ij}^{\left(q\right)}=\left\{a_{ij}^{\left(q\right)}\right\}$ ; $q=\mathrm{1,2}$ and $A=\left[\begin{array}{cc}{A}^{\left(1\right)}& A^{\left(2\right)}\end{array}\right]$

$$A=n\times \left[\begin{array}{cc}\begin{array}{cc}{a}_{11}^{\left(1\right)}& a_{12}^{\left(1\right)}\\ a_{21}^{\left(1\right)}& a_{22}^{\left(1\right)}\end{array}& \begin{array}{cc}{a}_{11}^{\left(2\right)}& a_{12}^{\left(2\right)}\\ a_{21}^{\left(2\right)}& a_{22}^{\left(2\right)}\end{array}\end{array}\right]$$

Upon using Cordeiro and Klein (1994) modification of the Cox and Snell (1968) result; the $Bias\left(\begin{array}{c}\widehat{\alpha }\\ \widehat{\beta }\end{array}\right)=K^{-1}A vec \left(K^{-1}\right)$The bias-adjusted estimators can be obtained as in equation (32)

$$\left(\begin{array}{c}{\alpha }^{\ast }\\ {\beta }^{\ast }\end{array}\right)={\widehat{K}}^{-1}\widehat{A} vec \left({\widehat{K}}^{-1}\right)$$

(32)

Where $\widehat{K}=K{|}_{\alpha =\widehat{\alpha }\& \beta =\widehat{\beta }}$ and $\widehat{A}=A{|}_{\alpha =\widehat{\alpha }\& \beta =\widehat{\beta }}$

3. Section 3

3.1. Variance-Corrected MLE Procedure and MBUW

Maximum likelihood estimators for the parameters of MBUW distribution may have large variances or infinite variances. Because the parameters are correlated they also have large covariance. The surface of the log-likelihood function may exhibits a flat shape which is reflected by multiple pair of estimators that fit the distribution. Also the estimators may be unstable. Inspecting the surface of the negative log-likelihood function can depicts the pairs of parameters which attain the minimal negative likelihood (nLL) values. The approach used by the author in this paper is to obtain these pairs and find the relationship between them and define one parameter in term of the other. Then reparameterize the negative likelihood function and scale one of the parameter in log scale. Then estimate this parameter and back substitute in the relationship to obtain the other parameter. The variance of the parameter obtained from the MLE is obtained by inversing the fisher information while the parameter defined in the relationship is obtained using the delta method.

3.2. Real Data Analysis Using Variance-Corrected MLE

These data were mentioned by the author in previous work (Iman M. Attia, 2024)

Steps of the technique used by the author:

1-

Inspect the surface of the nLL.

2-

Extract the pairs of alpha and beta parameter that minimize the nLL. The range of the alpha and beta were divided into equally spaced 500 points. The range differs according to the datasets. These ranges were chosen according to the results obtained from MLE using Nelder Mead algorithm results.

3-

Define beta in terms of alpha by fitting the best curve for this relationship.

4-

Reparameterize the nLL using this relationship.

5-

Estimate alpha parameter.

6-

Back substitute in the relationship to obtain the beta parameter.

7-

Use inverse fisher obtained from the MLE to obtain variance of the alpha and use the delta method to obtain the variance of the beta.

8-

Use goodness of fit test like the KS-test, AD-test and CVM-test evaluate the fitting of the distribution to data.

3.2.1. First Dataset: Dwelling Without Basic Facilities

1-

Inspecting the surface of nLL. Figure 1 illustrates this surface of which has a flat part.

2-

Extracting the pairs of the parameters that minimize the nLL. These pairs were 130 pairs. Figure 2 illustrates the relation between these parameters.

3-

Define the relationship between alpha and beta by fitting the best curve depicting this relationship. see equations (1-3)

For exponentional decay model: $\beta =5.4726e^{-0.7058}+0.845\dots .\left(1\right)$

For polynomial model : $\beta =0.0618{\alpha }^2-0.7674\alpha +3.279\dots .\left(2\right)$

For reciprocal model: $\beta ={\displaystyle \frac{3.6607}{\alpha }}+0.2907\dots \dots .(3)$

4-

Reparameterize the nLL by substituting each beta in LL. see equation(4)

$$l\left(\alpha \right)=nln\left(6\right)-n\beta ln\left(\alpha \right)+\sum _{i=1}^{n}ln\left[1-y_i^{{\displaystyle \frac{1}{{\alpha }^{\beta }}}}\right]+\left({\displaystyle \frac{2}{{\alpha }^{\beta }}}-1\right)\sum _{i=1}^{n}ln\left(y_i\right)$$

(4)

5-

Use the Nelder Mead algorithm to obtain MLE estimator for the alpha and use the inverse fisher to obtain the variance of the alpha.

6-

Back substitute in each model to obtain the beta.

6-

Use the delta method to obtain the variance of the beta as follows: delta method is defined as $var\left(g\left(\widehat{\theta }\right)\right)={\left(g^{\text{'}}\left(\theta \right)\right)}^{2}var\left(\widehat{\theta }\right)$ see equations(5-8)

For the three models it is defined as: $var\left(\widehat{\beta }\right)={\left[{\displaystyle \frac{d}{d\alpha }}\left(\beta \right)\right]}^{2}var\left(\widehat{\alpha }\right)\dots (5)$

For the exponen. Decay : ${\displaystyle \frac{d}{d\alpha }}\beta =5.4726\left(-0.7058\right)e^{-0.7058\alpha }\dots ..(6)$

For the quadratic polynomial: ${\displaystyle \frac{d}{d\alpha }}\beta =2\mathrm{*}0.0618\alpha -0.7674\dots (7)$

For the reciprocal model ${\displaystyle \frac{d}{d\alpha }}\beta ={\displaystyle \frac{-3.6607}{{\alpha }^2}}\dots .(8)$

8-

Use goodness of fit techniques such as KS-test to test the null hypothesis testing the fitness of the data to the theoretical distribution with the estimated parameters.

The results of the above procedure is shown in Table 1

The exponential decay has the following metrics: residual sum of square (SRR) value 0.0493, R sqaure value 0.9961, and root mean square error (RMSE) value 0.0195. The reciprocal model has the following values for RSS, R squared and, RMSE: 0.1198, 0.9905 and, 0.0305 respectively. The quadratic model has the following values for RSS, R squared and, RMSE: 0.2442, 0.9807 and 0.0435 respectively.

The analysis revealed the marked reduction in the variance for both alpha and beta. This has tremendous effect when constructing the confidence interval (CI) as it becomes narrower. The values of the parameters were not unique; they show small changes when changing the initial guess. The values with minimum variance were chosen to be reported as long as they failed to reject the null hypothesis. The other metrics like nLL, AIC, corr AIC, BIC, and HQIC are the same as the one that are obtained from conducting Nelder mead algorithm to estimate MLE for both parameters. And this is reflected on the graph of the theoretical CDF, graph of PDF and the QQ plot. They are all the same as if there is the same pattern if the parameters give the same nLL. The values of alpha and beta are nearly equal and the variances show minor differences among the models but the quadratic model has the minimal variance.

3.2.2. Second Dataset: Quality of the Support Network

1-

Inspecting the surface of nLL. Figure 3 illustrates this surface of which has a flat part.

2-

Extracting the pairs of the parameters that minimize the nLL. These pairs were 16 pairs. Figure 4 illustrates the relation between these parameters.

3-

Define the relationship between alpha and beta by fitting the best curve depicting this relationship. See equation (9)

For linear model: $\beta =4.2007\alpha +0.442\dots \dots .(9)$

Follow the steps from 4 to 8 and Table 2 and Table 3 shows the results.

The linear model has the following values for RSS, R squared and, RMSE: 0.0029, 0.9977 and, 0.014 respectively.

The variance of both parameters show marked reduction. The confidence (CI) interval becomes narrower. The distribution fits the data. Other metrics are more or less the same as the results obtained from MLE using the Nelder Mead algorithm. The parameters do not change with the initial guess.

3.2.3. Third Dataset: Voter Turnout Dataset

1-

Inspecting the surface of nLL. Figure 5 illustrates this surface of which has a flat part.

2-

Extracting the pairs of the parameters that minimize the nLL. These pairs were 51 pairs. Figure 6 illustrates the relation between these parameters.

3-

Define the relationship between alpha and beta by fitting the best curve depicting this relationship. See equation(10-11)

For quadratic model: $\beta =4.3363{\alpha }^2-0.86455\alpha +0.47298\dots .(10)$

For the power law model: $\beta =3.8879{\alpha }^{2.4616}+0.4082\dots .(11)$

Follow the steps from 4 to 8, for the delta method see equation (12-13):

For the power law: ${\displaystyle \frac{d}{d\alpha }}\beta =3.8879\left(2.4616\right){\alpha }^{1.4616}\dots (12)$

For the quadratic polynomial: ${\displaystyle \frac{d}{d\alpha }}\beta =2\mathrm{*}4.3363\alpha -0.8645\dots (13)$

The power law model has the following values for RSS, R squared and RMSE: 0.0289, 0.997, and 0.024 respectively. The quadratic model has the following values for RSS, R squared and RMSE: 0.0422, 0.9956, and 0.0291 respectively.

Figure 5. negative likelihood surface using the Voter dataset.

Figure 5. negative likelihood surface using the Voter dataset.

Figure 6. Nonlinear models for the voter dataset (increasing convex relationship between alpha and the beta), with residual sum of squares (RSS), R^2 and root of mean square error(RMSE) illustrated in the figure for each curve, high lightening that the power law is the best model followed by the polynomial( quadratic).

Figure 6. Nonlinear models for the voter dataset (increasing convex relationship between alpha and the beta), with residual sum of squares (RSS), R^2 and root of mean square error(RMSE) illustrated in the figure for each curve, high lightening that the power law is the best model followed by the polynomial( quadratic).

Table 4 illustrates the results of using these models to re-parameterize the nLL and running MLE using the Nelder Mead algorithm by Matlab.

From Table 4, the analysis revealed that the metrics between the two models are minor but it is in favor of the polynomial model over the power law model. This is due to the fact that it has more negative values of the nLL, AIC, corr AIC, BIC, HQIC than the power law has. It also has less value for the KS-test, AD and CVM than the power law has. All these metrics favor the quadratic model over the power law model, although the power law model better describes the relationship between beta and alpha than the quadratic model as indicated by the metrics; SRR, R squared and RMSE.

3.2.4. Fourth Dataset: Flood Data

1-

Inspecting the surface of nLL. Figure 7 illustrates this surface of which has a flat part.

2-

Extracting the pairs of the parameters that minimize the nLL. These pairs were 139 pairs. Figure 8 illustrates the relation between these parameters.

3-

Define the relationship between alpha and beta by fitting the best curve depicting this relationship. See equations(14-16)

For exponentional decay model: $\beta =19361e^{-9.1127}+0.1435\dots .(14)$

For polynomial model: $\beta =0.889{\alpha }^2-3.5278\alpha +3.5557\dots \dots .(15)$

For reciprocal model: $\beta ={\displaystyle \frac{1.3445}{\alpha }}-0.5694\dots \dots .(16)$

Follow the steps from 4 to 8

The exponential decay model has the following values for RSS, R squared and, RMSE: 0.2612, 0.969, and 0.0435 respectively. The quadratic model has the following values for RSS, R squared and, RMSE: 2.1792, 0.7412 and 0.1257 respectively. The reciprocal model has the following values for RSS, R squared and, RMSE: 2.9594, 0.6485 and, 0.1464 respectively.

Table 5 illustrates the results obtained from re-parameterizing the nLL function using these 3 different models.

In the analysis, initial guess may cause the estimators to change their values, but the variance and the goodness of fit may judge or add to the control if this initial guess, causing the new value, should be taking into consideration or discarded. In this situation the low variance estimator associated with goodness of fit test supporting the fitting distribution are the corner stones to choose the estimators values and hence consider the stability of the estimators.

The variance of alpha obtained by exponential decay model is the lowest than other variances obtained by quadratic and reciprocal model. The opposite is true concerning the variance of the beta. The variances of both alpha and beta for the reciprocal model are nearly equal as well as the standard errors. This is reflected by a narrower CI for both alpha and beta estimators obtained from the reciprocal model. This is not true concerning the interval for the beta estimator obtained from exponential decay model which is the widest among the other CI intervals obtained from other models. So the reciprocal model may outperform the exponential model

3.2.5. Fifth Dataset: Time Between Failures Data Set

• Inspecting the surface of nLL. Figure 9 illustrates this surface of which has a flat part.
• Extracting the pairs of the parameters that minimize the nLL. These pairs were 246 pairs. Figure 10 illustrates the relation between these parameters.
• Define the relationship between alpha and beta by fitting the best curve depicting this relationship. See equations (17-19)

For exponential decay model: $\beta =7.1594e^{-0.97948}+0.6735\dots .(17)$

For polynomial model: $\beta =0.1011{\alpha }^2-1.0218\alpha +3.2653\dots \dots .(18)$

For reciprocal model: $\beta ={\displaystyle \frac{3.1318}{\alpha }}+0.0648\dots \dots .(19)$

Follow the steps from 4 to 8

The exponential decay model has the following values for RSS, R squared and, RMSE: 0.1239, 0.9957, and 0.0225 respectively. The reciprocal model has the following values for RSS, R squared and, RMSE: 0.5583, 0.9808 and, 0.0477 respectively. The quadratic model has the following values for RSS, R squared and, RMSE: 0.9653, 0.9668 and 0.0628 respectively.

Table 6 illustrates the results obtained from re-parameterizing the nLL function using these 3 different models.

The analysis of the above models are comparable the variances are approximately equal, the estimator are also nearly equal. Any model can be chosen. The CI is narrow. The estimators are stable regardless the initial guess used as the estimator, variance and the goodness of fit results does not change. This is peculiar for this dataset.

3.2.6. Sixth Dataset: Capacity Factor Data Set

• Inspecting the surface of nLL. Figure 11 illustrates this surface of which has a flat part.
• Extracting the pairs of the parameters that minimize the nLL. These pairs were 80 pairs. Figure 22 illustrates the relation between these parameters.
• Define the relationship between alpha and beta by fitting the best curve depicting this relationship. See equation (20-22)

For exponential decay model: $\beta =9.424e^{-1.2276}+0.6253\dots .(20)$

For polynomial model: $\beta =0.1596{\alpha }^2-1.3633\alpha +3.537\dots .(21)$

For reciprocal model: $\beta ={\displaystyle \frac{3.0101}{\alpha }}-0.0811\dots \dots .(22)$

Follow the steps from 4 to 8

The exponential decay model has the following values for RSS, R squared and, RMSE: 0.0442, 0.9954, and 0.0237 respectively. The reciprocal model has the following values for RSS, R squared and, RMSE: 0.2843, 0.9701 and, 0.06 respectively. The quadratic model has the following values for RSS, R squared and, RMSE: 0.3523, 0.963 and 0.0668 respectively.

Table 7 illustrates the results obtained from re-parameterizing the nLL function using these 3 different models.

The results of the three models are comparable. The variances of parameters are small and are approximately equal between the 3 models. The CI is small. Any model can be chosen.