Generalized Bayesian Inference Study Based on Type-II Censored Data from the Class of Exponential Distributions

1. Introduction

In the Bayesian inference techniques, GB analysis was introduced and studied based on the learning rate parameter ($0<\eta <1$). The traditional Bayesian framework for $\eta =1$ is a fraction of the power of the likelihood function $L\left(\theta \right)\equiv L(\theta ;data)$ for the parameter $\theta \in \mathsf{\Theta }$. This means that if the prior distribution of the parameter $\theta$ is $\pi \left(\theta \right)$, then the GB posterior distribution for $\theta$ is

$${\pi }^{\mathrm{*}}\left(\theta \right| data)\propto L^{\eta }(\theta \left)\pi \right(\theta ), \theta \in \mathsf{\Theta }, 0<\eta <1.$$

(1)

For more information on the GB approach and how to select the value for the rate parameter, see [1,2,3,4,5,6,7,8,9,10,11,12,13]. Specifically, the choice of the learning rate was studied in [3]- [6] using the Safe Bayes algorithm based on the minimization of a sequential risk measure. In [7] and [8], another learning rate selection method was proposed, which included two different information adaptation strategies. The authors in [12] investigated GBE based on a joint type-II censored sample from multiple exponential populations, using various values of the learning rate parameter. The same study was presented in [13] but was based on joint hybrid censoring.

A one-sample prediction scheme is a Bayesian prediction method that determines the point predictor or prediction interval for unknown future values in the same sample based on the currently available observations. A two-sample prediction scheme or a multiple-sample prediction scheme are two other ways in which Bayesian prediction can utilize currently available observations to predict one or more future samples. Numerous authors have addressed the prediction of future failures or samples using different censoring techniques in the context of different prediction methods. We highlight some points that are relevant to our research. For instance, [13] investigated the GBP using a combined type-II censored sample drawn from multiple exponential populations. A study using a joint type-II censored sample from two exponential populations for Bayes estimation and prediction was published in [14]. Based on a generalized order statistic and multiple type II censoring, a Bayesian prediction for the future values of distributions from the class of exponential distributions was constructed in [15,16].

In the Bayesian study, the parameter of the distribution under investigation is a random variable, i.e. this unknown parameter is distributed according to the prior distribution. Empirical Bayes (EB) is a Bayesian study in which the parameters of the prior distribution (hyperparameters) are also unknown. By combining the density function of the distribution and the prior distributions, we obtain the marginal density function of the hyperparameters, which is used to estimate the hyperparameters. Therefore, the data of the original distribution are used to find the maximum likelihood estimators (MLEs) of these hyperparameters. EB has been introduced by many authors, for example, [17] studied the EBE of reliability performances with progressive type-II censoring of the Lomax model. The reliability and hazard function of the Kumaraswamy distribution were determined by [18] using progressive censored type II samples to estimate the EBE of the parameters. The Rayleigh distribution was studied in [19] to determine EBE and EBP.

In this study, the GBE, GEBE, GBP and GEBP distributions from the class of exponential distributions were examined using type II censored samples. Thus, the aim of this study is to examine all GB and GEB results for different LRP values, including $\eta =1$, which discribe the traditional Bayes.

The rest of this article is organized as follows. Section 2 introduces the class of exponential models and then describes the problem of GB, GEB, GBP and GEBP for this class. Section 3 applies the investigation from Section 2 to the exponential and Rayleigh models, which are given as examples of the class. In addition, in that section, we present the simulation study for the exponential and Rayleigh models to obtain the GBE, GEBE, GBP and GEBP for different LRP values and compare the results. Finally, Section 4 discusses the results and concludes the paper.

2. Estimation and Prediction

In this section, we introduce the exponential class of models and examine the problems of the GB, GEB, GBP, and GEBP for this class.

2.1. The Model

Let $\mathit{\theta }$ be the vector of parameters, define a function $g\left(x;\mathit{\theta }\right)\equiv g\left(x\right)$, and its derivative $g^{\prime }\left(x\right)$ where $\underset{x\to \infty }{\lim }g\left(x\right)=\infty , \underset{x\to 0^{+}}{\lim }g\left(x\right)=0.$The probability density function (pdf), the cumulative probability density function (cdf) and the survival function (sf) of the exponential class are each given by:

$$f\left(x;\mathit{\theta }\right)=g^{\prime }\left(x\right)\exp \left[-g\left(x\right)\right], x>0, \mathit{\theta }>0;$$

(1)

$$F\left(x;\mathit{\theta }\right)=1-\exp \left[-g\left(x\right)\right];$$

(2)

and

$$\overline{F}\left(x;\mathit{\theta }\right)=\exp \left[-g\left(x\right)\right].$$

(3)

The likelihood function under type-II censored data from the class is given by,

$$\begin{gathered} L\left(\underset{\_}{x};\mathit{\theta }\right)=c {\overline{F}\left(x_r;\mathit{\theta }\right)}^{n-r}{\prod }_{i=1}^r\mathrm{‍}f\left(x_i;\mathit{\theta }\right) \\ \propto A\left(\underset{\_}{x};\mathit{\theta }\right)\exp \left[-B\left(\underset{\_}{x};\mathit{\theta }\right)\right], \end{gathered}$$

(4)

where, $c={\displaystyle \frac{n!}{\left(n-r\right)!}}, A\left(\underset{\_}{x};\mathit{\theta }\right)=\prod _{i=1}^r\mathrm{‍}{g}^{\prime }\left(x_i\right),B\left(\underset{\_}{x};\mathit{\theta }\right)={\sum }_{i=1}^r\mathrm{‍}g\left(x_i\right)+\left(n-r\right)g\left(x_r\right),$

$$\underset{\_}{x}=\left(x_1,\dots ,x_r\right)$$

Consider the prior distribution of $\mathit{\theta }$ in the following general form,

$$\pi (\mathit{\theta };\mathit{\delta })={I_{\mathit{\delta }}}^{-1}C\left(\mathit{\theta };\mathit{\delta }\right),$$

(5)

where, $I_{\mathit{\delta }}={\int }_{\mathit{\theta }}C\left(\mathit{\theta };\mathit{\delta }\right)\exp \left[-D\left(\mathit{\theta };\mathit{\delta }\right)\right]d\mathit{\theta }$, $\mathit{\delta }$ is a vector of hyperparameters.

Combining (4) and (5), after raising (4) to the fractional power $\eta$, the GB posterior distribution of $\mathit{\theta }$ is given by,

$$\begin{gathered} {\pi }_G^{*}\left(\mathit{\theta };\mathit{\delta },\underset{\_}{x}\right)={I_{\mathit{\delta }}^{*}}^{-1}{L}^{\eta }\left(\underset{\_}{x};\mathit{\theta }\right)\pi \left(\mathit{\theta };\mathit{\delta }\right) \\ ={I_{\mathit{\delta }}^{*}}^{-1} G\left(\mathit{\theta };\mathit{\delta },\underset{\_}{x}\right)\exp \left[-H\left(\mathit{\theta };\mathit{\delta },\underset{\_}{x}\right)\right] \end{gathered}$$

(6)

where $G_{\mathit{\delta }}=A^{\eta }\left(\underset{\_}{x};\mathit{\theta }\right)C\left(\mathit{\theta };\mathit{\delta }\right),H_{\mathit{\delta }}=\eta B\left(\underset{\_}{x};\mathit{\theta }\right)+D\left(\mathit{\theta };\mathit{\delta }\right),$ and $I_{\mathit{\delta }}^{*}={\int }_{\mathit{\theta }}{G}_{\mathit{\delta }}\exp \left[-H_{\mathit{\delta }}\right]d\mathit{\theta }.$

Then GBE is given by,

$${\widehat{\mathit{\theta }}}_{GB}=E\left(\mathit{\theta }\right)={\int }_{\mathit{\theta }}\mathit{\theta }{ \pi }_G^{*}\left(\mathit{\theta };\mathit{\delta },\underset{\_}{x}\right) d\mathit{\theta }.$$

(7)

2.2. Generalized empirical Bayesian estimation

Combining (1) and (5), to get the marginal pdf $f\left(x;\mathit{\delta }\right)$ as follows

$$\begin{gathered} f\left(x;\mathit{\delta }\right)={\int }_{\mathit{\theta }}\pi \left(\mathit{\theta };\mathit{\delta }\right)f\left(x;\mathit{\theta }\right)d\mathit{\theta } \\ ={I_{\mathit{\delta }}}^{-1}{\int }_{\mathit{\theta }}C\left(\mathit{\theta };\mathit{\delta }\right)g^{\prime }\left(x\right)\exp \left[-\left\{D\left(\mathit{\theta };\mathit{\delta }\right)+g\left(x\right)\right\}\right]d\mathit{\theta }. \end{gathered}$$

(8)

From pdf in (8) we get the cdf $F\left(x;\mathit{\delta }\right)$, then the likelihood function under type-II censored data is given by,

$$L_E\left(\underset{\_}{x};\mathit{\delta }\right)=c {\overline{F}\left(x_r;\mathit{\delta }\right)}^{n-r}{\prod }_{i=1}^r\mathrm{‍}f\left(x_i;\mathit{\delta }\right),$$

(9)

Using the loglikelihood function ${\mathcal{L}}_E\left(\underset{\_}{x};\mathit{\delta }\right)=\log L_E\left(\underset{\_}{x};\mathit{\delta }\right)$, to find the maximum likelihood estimator (MLE) $\widehat{\mathit{\delta }}$ as follows

$$\widehat{\mathit{\delta }}=\underset{\mathit{\delta }}{\mathrm{argmax}}{\mathcal{L}}_E\left(\underset{\_}{x};\mathit{\delta }\right).$$

(10)

By solving the following equation,

$${\displaystyle \frac{\partial {\mathcal{L}}_E\left(\underset{\_}{x};\mathit{\delta }\right)}{\partial \mathit{\delta }}}=0.$$

(11)

Substituting by $\widehat{\delta }$ in (6) we obtain the posterior GE as follows,

$${\pi }_{GE}^{*}\left(\mathit{\theta };\widehat{\mathit{\delta }},\underset{\_}{x}\right)={I_{\widehat{\mathit{\delta }}}^{*}}^{-1} G_{\widehat{\mathit{\delta }}}\exp \left[-H_{\widehat{\mathit{\delta }}}\right].$$

(12)

Then GEBE given by

$${\widehat{\mathit{\theta }}}_{GE}=E\left(\mathit{\theta }\right)={\int }_{\mathit{\theta }}\mathit{\theta }{\pi }_{GE}^{*}\left(\mathit{\theta };\widehat{\mathit{\delta }},\underset{\_}{x}\right) d\mathit{\theta }.$$

(13)

2.3. A one sample prediction scheme

To determine the GBP and GEBP intervals using a one-sample prediction scheme under the type-II censored sample from the class, the first $r$ordered statistics $\underset{\_}{x}$are observed from a random sample of size $n; r<n .$A one sample prediction scheme is considered to predict the rest of unobserved values $x_s, s=r+1, \dots , n.$ The conditional density function of $x_s$ given $\underset{\_}{x}$ is given by

$$f\left(x_s|\underset{\_}{x}\right)={\displaystyle \frac{\left(n-r\right)!}{\left(s-r-1\right)!\left(n-s\right)!}} {\left[\overline{F}\left(x_r\right)-\overline{F}\left(x_s\right)\right]}^{s-r-1} {\overline{F}\left(x_s\right)}^{n-s} {\overline{F}\left(x_r\right)}^{-(n-r)}f\left(x_s\right).$$

(14)

Substituting by (1), (3) in (14), the conditional density function of $x_s$ given $\underset{\_}{x}$ is,

$$f\left(x_s|\underset{\_}{x}\right)={\sum }_{j=0}^{s-r-1}{c}_j{g}^{\prime }\left(x_s\right)\exp \left[-n_j\left\{g\left(x_s\right)-g\left(x_r\right)\right\}\right],$$

(15)

where, $c_j={\displaystyle \frac{{\left(-1\right)}^{s-r-j-1} \left(n-r\right)!}{j!\left(s-r-j-1\right)!\left(n-s\right)!}} , n_j=n-r-j.$Combining (6) and (15) then integrating with respect to $\mathit{\theta }$, the GB predictive density function is given by,

$$f_G^{*}\left(x_s|\underset{\_}{x}\right)={I_{\mathit{\delta }}^{*}}^{-1}{\sum }_{j=0}^{s-r-1}{c}_j{\int }_{\mathit{\theta }}{g}^{\prime }\left(x_s\right)G_{\mathit{\delta }}\exp \left[-\left\{n_j\left(g\left(x_s\right)-g\left(x_r\right)\right)+H_{\mathit{\delta }}\right\}\right]d\mathit{\theta }.$$

(16)

The predictive reliability function of${ x}_s, s=r+1, \dots , n$ is given by,

$$\left(t|\underset{\_}{x}\right)={I_{\mathit{\delta }}^{*}}^{-1}{\sum }_{j=0}^{s-r-1}{\displaystyle \frac{c_j}{n_j}}{\int }_{\mathit{\theta }}{G}_{\mathit{\delta }}\exp \left[-\left\{n_j\left(g\left(t\right)-g\left(x_r\right)\right)+H_{\mathit{\delta }}\right\}\right]d\mathit{\theta }.$$

(17)

Equating (17) to ${\displaystyle \frac{1+\alpha }{2}} \mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{1-\alpha }{2}}$, respectively, we obtain ($1-\alpha )\%$ GBP bounds ${(L}_{\mathit{\delta }},U_{\mathit{\delta }})$. GEBP bounds ${(L}_{\widehat{\delta }},U_{\widehat{\delta }})$, can be obtained by substituting by $\widehat{\delta }$ in (17) then equating to ${\displaystyle \frac{1+\alpha }{2}} \mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{1-\alpha }{2}}$.

3. Applications

In this section, we apply the results in the previous section to one parameter models, therefore the models that discussed here are exponential $\mathrm{E}\mathrm{x}\mathrm{p}\left(\theta \right)$ and Rayliegh $\mathrm{R}\mathrm{a}\mathrm{y}\left(\theta \right)$. For these two distributions, the parameter $\theta$ assumed to be unknown, we may consider the conjugate prior distribution of $\theta$ as gamma prior distribution, $\theta \sim Gam({\delta }_1,{\delta }_2)$, hence,

$$C\left(\theta ;\mathit{\delta }\right)={\theta }^{{\delta }_1-1},D\left(\theta ;\mathit{\delta }\right)={\delta }_2\theta ; I_{\mathit{\delta }}^{-1}={\displaystyle \frac{{{\delta }_2}^{{\delta }_1}}{\mathsf{\Gamma }\left({\delta }_1\right)}} , {\delta }_1, {\delta }_2>0$$

(18)

3.1. Exponential model

Here we give the essential functions and important forms derived in Section 2 for the exponential model as follows:

$$g\left(x\right)=\theta x, x>0$$

(19)

For the likelihood function we have

$$A\left(\underset{\_}{x};\theta \right)={\theta }^r, T_E={\sum }_{i=1}^r\mathrm{‍}{x}_i+\left(n-r\right)x_r \mathrm{and} B\left(\underset{\_}{x};\theta \right)=\theta T_E.$$

Generalized posterior function, can be formed from the following,

$$G_{\mathit{\delta }}={\theta }^{\eta r{+\delta }_1-1}, H_{\mathit{\delta }}=\theta (\eta T_E+{\delta }_2) \mathrm{and} I_{\mathit{\delta }}^{*}={\displaystyle \frac{\mathsf{\Gamma }\left(\eta r{+\delta }_1\right)}{{(\eta T_E+{\delta }_2)}^{\eta r{+\delta }_1.}}}$$

The GBE of the parameter $\theta$ is given by,

$${\widehat{\theta }}_{GB}={\displaystyle \frac{\eta r{+\delta }_1 }{\eta T_E+{\delta }_2}}$$

(20)

The predictive reliability function of $x_s$ is given by,

$${\overline{F}}_{\mathit{\delta }}\left(t|\underset{\_}{x}\right)={\sum }_{j=0}^{s-r-1}{\displaystyle \frac{c_j}{n_j}}{[1+{\displaystyle \frac{n_j\left(t-x_r\right)}{\eta T_E+{\delta }_2}}]}^{-\left(\eta r{+\delta }_1\right)}.$$

(21)

Equating (21) to ${\displaystyle \frac{1+\alpha }{2}} \mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{1-\alpha }{2}}$, respectively, we obtain ($1-\alpha )\%$ GBP bounds ${(L}_{\mathit{\delta }},U_{\mathit{\delta }})$.

The functions and forms under GEB study can be illustrated as follows:

The marginal pdf $f\left(x;\mathit{\delta }\right)$ is given in the following form,

$$f\left(x;\mathit{\delta }\right)={\displaystyle \frac{{\delta }_1{{\delta }_2}^{{\delta }_1}}{{\left(x+{\delta }_2\right)}^{\left({\delta }_1+1\right)}}}$$

(22)

Using pdf $f\left(x;\mathit{\delta }\right)$ and cdf $F\left(x;\mathit{\delta }\right)$we get the likelihood function based on type-II censored data, as follows,

$$L_{\mathit{E}}\left(\underset{\_}{\mathit{x}};\mathit{\delta }\right)\propto {\displaystyle \frac{{ \left({\delta }_1{{\delta }_2}^{{\delta }_1}\right)}^r\prod _{i=1}^r{\left(x_i+{\delta }_2\right)}^{-\left({\delta }_1+1\right)}\mathrm{‍}}{{\left(1+\frac{x_r}{{\delta }_2}\right)}^{{\left(n-r\right)\delta }_1}}}.$$

(23)

By differentiating the loglikelihood function ${\mathcal{L}}_E\left(\underset{\_}{x};\mathit{\delta }\right)$ w. r. to ${\delta }_1{\mathrm{a}\mathrm{n}\mathrm{d}\delta }_2$ and equating each equation to zero, then solving them numerically we get the estimators ${\widehat{\delta }}_1 \mathrm{a}\mathrm{n}\mathrm{d} {\widehat{\delta }}_2$.

The GEBE of the parameter $\theta$ is given by,

$${\widehat{\theta }}_{GE}={\displaystyle \frac{\eta r+{\widehat{\delta }}_1}{\eta T_E+{\widehat{\delta }}_2}}$$

(24)

The predictive reliability function of $x_s$ is given by,

$${\overline{F}}_{\widehat{\mathit{\delta }}}\left(t|\underset{\_}{x}\right)=\sum _{j=0}^{s-r-1}{\displaystyle \frac{c_j}{n_j}} {\left[1+{\displaystyle \frac{n_j\left(t-x_r\right)}{\eta T_E+{\widehat{\delta }}_2}}\right]}^{-(\eta r+{\widehat{\delta }}_1)}.$$

(25)

Equating (25) to $(1+\alpha )/2 \mathrm{a}\mathrm{n}\mathrm{d}(1-\alpha )/2$, respectively, we obtain ($1-\alpha )\%$ GEBP bounds ${(L}_{\widehat{\delta }},U_{\widehat{\delta }})$.

3.2. Rayleigh model

The essential functions and important forms derived in Section. 2 and Section. 3 for Rayliegh distribution are derived as follows:

$$g\left(x\right)={\displaystyle \frac{\theta x^2}{2}}, x>0$$

(26)

For the likelihood function we have

$$A\left(\underset{\_}{x};\theta \right)={\theta }^r{\prod }_{i=1}^r\mathrm{‍}{x}_i, T_R={\sum }_{i=1}^r\mathrm{‍}{x}_i^2/2+\left(n-r\right)x_r^2/2 \mathrm{and} B\left(\underset{\_}{x};\theta \right)=\theta T_R.$$

Generalized posterior function, can be formed from the following,

$$G_{\mathit{\delta }}={\theta }^{\eta r{+\delta }_1-1}{\left({\prod }_{i=1}^r\mathrm{‍}{x}_i\right)}^{\eta }, H_{\mathit{\delta }}=\theta (\eta T_R+{\delta }_2) \mathrm{and} I_{\mathit{\delta }}^{*}={\displaystyle \frac{{\left({\prod }_{i=1}^r\mathrm{‍}{x}_i\right)}^{\eta }\mathsf{\Gamma }\left(\eta r{+\delta }_1\right)}{{(\eta T_R+{\delta }_2)}^{\eta r{+\delta }_1}}}.$$

The GBE of the parameter $\theta$ is given by,

$${\widehat{\theta }}_{GB}={\displaystyle \frac{\eta r{+\delta }_1 }{\eta T_R+{\delta }_2}}.$$

(27)

The predictive reliability function of $x_s$ is given by,

$${\overline{F}}_{\mathit{\delta }}\left(t|\underset{\_}{x}\right)={\sum }_{j=0}^{s-r-1}{\displaystyle \frac{c_j}{n_j}}{[1+{\displaystyle \frac{n_j\left(t^2-x_r^2\right)/2}{\eta T_R+{\delta }_2}}]}^{-\left(\eta r{+\delta }_1\right)}.$$

(28)

Equating (28) to ${\displaystyle \frac{1+\alpha }{2}} \mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{1-\alpha }{2}}$, respectively, we obtain ($1-\alpha )\%$ GBP bounds ${(L}_{\mathit{\delta }},U_{\mathit{\delta }})$.

The marginal pdf $f\left(x;\mathit{\delta }\right)$ is given in the following form,

$$f\left(x;\mathit{\delta }\right)={\displaystyle \frac{{\delta }_1{{\delta }_2}^{{\delta }_1}}{{\left(x+{\delta }_2\right)}^{\left({\delta }_1+1\right)}}} .$$

(29)

Using pdf $f\left(x;\mathit{\delta }\right)$ and cdf $F\left(x;\mathit{\delta }\right)$we get the likelihood function based on type-II censored data, as follows:

$$L_{\mathit{E}}\left(\underset{\_}{\mathit{x}};\mathit{\delta }\right)\propto {\displaystyle \frac{{ \left({\delta }_1{{\delta }_2}^{{\delta }_1}\right)}^r\prod _{i=1}^r{\left(x_i+{\delta }_2\right)}^{-\left({\delta }_1+1\right)}\mathrm{‍}}{{\left(1+x_r/{\delta }_2\right)}^{{(n-r)\delta }_1}}}$$

(30)

By differentiating the loglikelihood function ${\mathcal{L}}_E\left(\underset{\_}{x};\mathit{\delta }\right)$ with respect to ${\delta }_1 {\mathrm{a}\mathrm{n}\mathrm{d} \delta }_2$ and equating each equation to zero, then solving them numerically we get the estimators ${\widehat{\delta }}_1 \mathrm{a}\mathrm{n}\mathrm{d} {\widehat{\delta }}_2$.

The GEBE of the parameter $\theta$ is given by,

$${\widehat{\theta }}_{GE}={\displaystyle \frac{\eta r+{\widehat{\delta }}_1}{\eta T_E+{\widehat{\delta }}_2}}$$

(31)

The predictive reliability function of $x_s$ is given by,

$${\overline{F}}_{\widehat{\mathit{\delta }}}\left(t|\underset{\_}{x}\right)={\sum }_{j=0}^{s-r-1}{\displaystyle \frac{c_j}{n_j}}{[1+{\displaystyle \frac{n_j\left(t^2-x_r^2\right)/2}{\eta T_R+{\widehat{\delta }}_2}}]}^{-(\eta r+{\widehat{\delta }}_1)}.$$

(32)

Equating (31) to ${\displaystyle \frac{1+\alpha }{2}} \mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{1-\alpha }{2}}$, respectively, we obtain ($1-\alpha )\%$ GEBP bounds ${(L}_{\widehat{\delta }},U_{\widehat{\delta }})$.

3.3. Numerical analysis

In this subsection, the results of the Monte Carlo simulation are presented to evaluate the performance of the inference methods derived in the previous sections.

The simulation study is designed and carried out for the two models as follows:

• Generate one sample from each distribution with size $n=50$, and choosing $r=30, 40, 50$.
• Based on the chosen values of the hyperparameters $\left({\delta }_1,{\delta }_2\right)=\left(4,2\right),$ the suggested value for the parameter is $\theta =2$, $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\theta$is obtained as the mean of gamma distribution in (18).
• For EB, we use MLE ${(\widehat{\delta }}_1,{\widehat{\delta }}_2)$ to compute ${\widehat{\theta }}_{GE}$, where the results MLE ${(\widehat{\delta }}_1,{\widehat{\delta }}_2)$ based on exponential and Rayliegh distributions are shown in Table 1.
• For the Monte Carlo simulations we use $M=\mathrm{10,000}$ replicates, therefore the estimator $\widehat{\theta }={\displaystyle \frac{\sum _{i=1}^M{\widehat{\theta }}_i}{M}}$, and the estimated risk, $ER=\sqrt{{\displaystyle \frac{\sum _{i=1}^M{({\theta }_{\mathrm{i}}-{\widehat{\theta }}_{\mathrm{i}})}^2}{M}}}.$
• Using (20), (24), (27) and (31), the estimation results are obtained and expressed by the estimator $\widehat{\theta }$ and ER for different values of LRP, where $\eta =0.1, 0.5, 1$.
• The results of GBE and GEBE for exponential and Rayliegh distributions are shown in Table 2 and Table 4.
• Prediction results are based on one sample from each distribution with size $n=20$, the number of observations is $r=15,$ we then compute the GBP, GEBP bounds and its lengths at $\alpha =0.05,$for the future values with $s=16, 18, 20$ using (28) and (32).
• The results of GBP and GEBP for exponential and Rayliegh distributions are shown in Table 3 and Table 5.

From Table 2. According to $\widehat{\theta }$ and ER, GBE becomes better for small value of LRP but for the large value of $\mathit{r}$, that means getting the best result at $\eta =0.1$ and $\mathit{r}=50$ (complete sample). GEBE becomes better for large value of LRP and for the large value of $\mathit{r}$, that means getting the best result at $\eta =1$ and $\mathit{r}=50.$ In general the result of GBE is better than that of GEBE.

From Table 3. According to the length of the interval, GBP and GEBP becomes better for large value of $\mathit{r} \mathrm{a}\mathrm{n}\mathrm{d}$ LRP, that means getting the best result at $\eta =1$ and $\mathit{r}=50$ (complete sample). In general, the result of GEBP is better than that of GBP.

From Table 4. GBE becomes better for small value of LRP but for the large value of $\mathit{r}$, that means getting the best result at $\eta =0.1$ and $\mathit{r}=50$ (complete sample). GEBE becomes better for large value of LRP and for the large value of $\mathit{r}$, except for the complete sample, the result becomes better for small value of LRP that means getting the best result at $\eta =0.1$ and $\mathit{r}=50.$ The result of GBE is better than that of GEBE at $\mathit{r}=30, 40$, but GEBE is better than GBE for the complete sample.

From Table 5. GBP and GEBP becomes better for large value of $\mathit{r} \mathrm{a}\mathrm{n}\mathrm{d}$ LRP, that means getting the best result at $\eta =1$ and $\mathit{r}=50$ (complete sample). In general, the result of GEBP is better than that of GBP.

4. Discussion and Conclusion

In this study, a one-parameter model belonging to the class of exponential models is considered. Two well-known models $Exp\left(\theta \right)$ and $Ray\left(\theta \right)$ are examined based on a censored type-II sample. GB, GEB, GBP and GEBP are discussed for the two distributions with different values of LRP $\eta$. From the results in Table 2 to Table 5, we can summarize the results of the two distributions as follows:

4.1. The result of exponential model

• GBE becomes better for small value of LRP but for the large value of $\mathit{r}$, that means getting the best result at $\eta =0.1$ and $\mathit{r}=50$. GEBE becomes better for large value of LRP and for the large value of $\mathit{r}$, that means getting the best result at $\eta =1$ and $\mathit{r}=50.$
• GBP and GEBP becomes better for large value of $\mathit{r} \mathrm{a}\mathrm{n}\mathrm{d}$ LRP, that means getting the best result at $\eta =1$ and $\mathit{r}=50$.
• The result of GBE is better than that of GEBE but the result of GEBP is better than that of GBP.
• Small values of LRP give the best result for GBE but vice versa for GEBP.

4.2. The result of Rayleigh model

• GBE becomes better for small value of LRP but for the large value of $\mathit{r}$, that means getting the best result at $\eta =0.1$ and $\mathit{r}=50$. GEBE becomes better for large value of LRP and for the large value of $\mathit{r}$, except for the complete sample the result becomes better for small value of LRP, that means getting the best result at $\eta =0.1$ and $\mathit{r}=50.$ The result of GBE is better than that of GEBE at $\mathit{r}=30, 40$, but GEBE is better than GBE for the complete sample.
• GBP and GEBP becomes better for large value of $\mathit{r} \mathrm{a}\mathrm{n}\mathrm{d}$ LRP, that means getting the best result at $\eta =1$ and $\mathit{r}=50$.
• The result of GBE is better than that of GEBE but the result of GEBP is better than that of GBP.
• Small values of LRP for the complete sample give the best result for GBE but vice versa for GEBP.