On estimation of augmented strength reliability parameters under non-informative priors: A non-identical case

Abstract

In this article we consider the Bayesian and Maximum Likelihood (ML) estimation of augmented strength of a system for the generalized case of the proposed Augmentation Strategy Plan (ASP). ASP has interesting applications in stress strength reliability. The Bayes estimation is performed by assuming non-informative (uniform and Jeffreys) types of priors under two different loss functions i.e. squared error loss function (SELF) and LINEX loss function (LLF) for better comprehension purpose. It is assumed that the strength ( $X$ ) and the imposed stress ( $Y$ ) follow independent and non-identical two parameter gamma distributions. The MCMC simulation techniques are employed to study the comparison between the ML and Bayes estimators of augmented strength reliability on the basis of their mean square errors (mse) and absolute biases. The proposed estimators are validated by Monte Carlo simulated as well as real data sets.

Keywords

Augmented strength reliability gamma distribution Bayes estimator non-informative prior Metropolis-Hasting algorithm Markov chain monte carlo simulation

1. Introduction

The two-parameter gamma distribution is a very popular distribution to analyze the life time data with its applications in reliability engineering and also in other fields of study e.g. life insurance claims, climatology, meteorology, telecommunication etc. (see Lawless 1982). In some of the certain conditions, the gamma distribution can be used as a counterpart of log-normal, weibull and similar type distributions to analyze any positive real data. This distribution possesses many interesting properties over exponential, weibull and log-normal distributions, one of them is called the reproductive property, which leads us to choose two parameter gamma distribution for the proposed ASP (see Section 2). The probability density function (pdf) of two parameter gamma distribution is given by

$f_{X}\left({x;\alpha,\lambda}\right)=\frac{\alpha^{\lambda}}{\Gamma\left(% \lambda\right)}x^{\lambda-1}\exp\left({-\alpha x}\right);x>0,\alpha,\lambda>0$ (1)

where, $\alpha$ and $\lambda$ are known as scale and shape parameters respectively and is denoted as $gamma(\alpha,\lambda)$ .

In this paper we consider the problem of Bayesian estimation of strength reliability $R=P\left({X>Y}\right)$ under the generalized case of ASP by assuming that both the strength ( $X$ ) and stress ( $Y$ ) have non-identical gamma distribution and are independent to each other. The problem of estimating $R=P\left({X>Y}\right)$ has attracted the researchers because of its applicability in various real life situations. The concept of stress-strength reliability was firstly introduced by Birnbaum (1956) and subsequently Birnbaum and McCarty (1958) and Govindarajulu (1968) discussed the procedure for finding distribution free confidence interval for stress-strength reliability. The pioneer works on the estimation of system reliability are Owen et al. (1964), Church and Harris (1970) and Downton (1973) who considered stress strength variables are normally distributed. Tong (1974), Sathe and Shah (1981) and Chao (1982) etc. have considered the estimation of system reliability by assuming that the stress and strength random variables are exponentially distributed.

A handful amount of works is available in the literature in context of system reliability problem for gamma distribution as stress-strength model. Constantine et al. (1986) attempted to find out the different representations of $R=P({X>Y})$ for real and integer valued shape parameters and the comparison between ML and uniformly minimum variance estimators of the stress-strength reliability were made by considering the shape parameters as known integer-valued. In addition, Constantine et al. (1989, 1990) extended their above cited work with non-parametric approach and bootstrap methods for finding the bootstrap and exact confidence interval of $R$ under gamma stress strength setup for integer valued shape parameters. Latter, Huang et al. (2012) consider the same models as Constantine et al. (1986) with relaxed assumption of known integer valued shape parameters and developed the system reliability problem for any known positive real valued shape parameters. They proposed ML and UMVU estimators of $R$ and a graphical representation of comparison was presented on the basis of their MSEs and biases for different values of known real valued shape parameters. An extensive amount of literature works is available towards the stress-strength reliability problems for both identical and non-identical choices of stress strength distributions. A detailed review on system reliability problem can be viewed in classical monograph of Kotz et al. (2003) and references therein.

Here, an attempt has been made to make a comparison between Bayesian and classical methods of estimation for the augmented strength reliability for the generalized case of ASP on the basis of their mean square errors and absolute biases. The proposed ASP consists three different possible cases to augment (enhance) the strength of weaker equipment. In many real life scenarios, there occur some early stage failures for a newly sophisticated system and also frequent failures occur in used or old systems due to continuous degradation of their strengths over time. Such types of failures are known as irrelevant failures. Hence, to overcome from such types of irrelevant failure problems the augmenting strength procedure may be suggestive. In this connection, the problem of augmentation of strength reliability and estimation of its parameters were attempted. Chandra and Sen (2014) suggested the name Augmentation Strategy Plan (ASP) for all the three possible cases of augmentation and they derived the augmented strength reliability model for gamma life time distribution. In similar fashion, Chandra and Rathaur (2015) studied for augmenting Inverse Gaussian stress strength reliability under ASP. Recently, Chandra and Rathaur (2016a, 2017a) have attempted Bayes estimation of augmenting gamma strength reliability of a System under non-informative as well as informative prior distributions respectively by assuming that the system strength and the common stress imposed on it are independently and identically distributed as gamma random variables. In this direction some more references can be viewed in Chandra and Rathaur (2016b, 2017b).

The rest of the article is organized as follows. In Section 2, generalized augmented strength reliability for non-identical stress strength under ASP is presented. In Section 3, the ML estimators of augmented strength reliability for different combinations of scale and shape parameters under ASP are presented. In Section 4, Bayes estimators of augmented strength reliability parameters for non-informative types of priors (uniform and Jeffrey’s priors) under SELF and LLF are considered. A real life and simulated data sets are analyzed to illustrate the proposed methods in Section 5. A simulation study and its discussion based on findings of the generalized case of ASP are reported in Section 6. Finally, the conclusion of the article is given in Section 7.

2. Generalized augmented strength reliability for non-identical gamma stress strength

Let $(X_{i};i=1,2,\ldots,n)$ be $n$ strength random variables which are independently and identically distributed as gamma distribution with parameters $(\alpha_{1},\lambda_{1})$ and $Y$ being the common stress encountered by the system distributed as gamma distribution with scale and shape parameters $\alpha_{2}$ and $\lambda_{2}$ respectively and it is also assumed that the strength and stress are independent to each other. To enhance the strength of weaker/failed equipment, we implemented the Augmentation Strategy Plan (ASP), which comprises three possible cases of ASP to enhance the strength reliability of an equipment to face the common stress proposed by Chandra and Sen (2014). The more generalized case (i.e. case III) of ASP can be stated as “The strength of the equipment is increased by adding ‘ $n$ ’ identical components each having strength $S_{i}=mX_{i}$ , which is ‘ $m$ ’ times of initial strength of the equipment, is set to face the common stress $(Y)$ of the equipment. Thus the augmented strength $Z_{3}=\sum_{i=1}^{n}{S_{i}}$ follow Gamma distribution with parameters $({{\alpha_{1}}/m},{n\lambda_{1}})$ and each $S_{i}(i=1,2,3,\ldots,n)$ follows gamma $({{\alpha_{1}}/m},{\lambda_{1}})$ independently”.

It is noticed that case-I and case-II of ASP were special cases of case-III, which we refer to as generalized case of ASP. The probability density function (pdf) of augmented strength $\left({Z_{k};k=1,2,3}\right)$ under generalized case of ASP can be given as

$f_{Z_{K}}\left({z_{k}/\alpha_{1},\lambda_{1}}\right)=\frac{1}{\Gamma\left({n% \lambda_{1}}\right)}\left({\frac{\alpha_{1}}{m}}\right)^{n\lambda_{1}}\exp% \left({-\frac{\alpha_{1}z_{k}}{m}}\right)z_{k}^{n\lambda_{1}-1};z_{k}>0,\alpha% _{1},\lambda_{1}>0,$ (2)

where, ‘ $m$ ’ is a positive real number and ‘ $n$ ’ is positive integer. The PDFs of case-I, case-II and case-III can directly be obtained by substituting $(k=n=1)$ , $(k=2,m=1)$ and $(k=3)$ respectively in Eq. 2. The system reliability parameter $R_{k}=P(Z_{k}>Y)$ , when the shape parameters are assumed to be real valued parameters, is given by

$R_{k}=\Pr(Z_{k}>Y)=P\left({\frac{Z_{k}}{Y}>1}\right)$

It may be noted that $\frac{2\alpha_{1}Z_{k}}{m}\sim\chi^{2}(2n\lambda_{1})$ and $2\alpha_{2}Y\sim\chi^{2}(2\lambda_{2})$ which are independently distributed. Thus the augmented strength reliability becomes

$\displaystyle R_{k}=P\left({\frac{{\{2({\alpha_{1}}/m)Z_{k}\}}/{(2n\lambda_{1}% )}}{{2\alpha_{2}Y}/{(2\lambda_{2})}}>\frac{\lambda_{2}}{n\lambda_{1}}\frac{1}{% \rho}}\right)=P\left({\tau>\frac{\lambda_{2}}{n\lambda_{1}}\frac{1}{\rho}}\right)$ $\displaystyle=1-F\left({\frac{\lambda_{2}}{\rho n\lambda_{1}};2n\lambda_{1},2% \lambda_{2}}\right)$ (3)

where, ${\rho=m\alpha_{2}}/{\alpha_{1}}$ and $F\left({.;2n\lambda_{1},2\lambda_{2}}\right)$ is cumulative distribution of F-distribution with $2n\lambda_{1}$ and $2\lambda_{2}$ degrees of freedom. Now suppose one of the shape parameters ( $\lambda_{1}$ , say) is considered to be an integer. Then the system reliability $R_{k}$ can be expressed as a finite sum, given as follows

$\displaystyle R_{k}=\Pr(Z_{k}>Y)=\int\limits_{0}^{\infty}{\left({\int\limits_{% 0}^{y}{f_{Z_{k}}\left({z_{k}}\right)}dz_{k}}\right)}g_{Y}(y)dy=\int\limits_{0}% ^{\infty}{F_{Z_{k}}(y)g_{Y}(y)dy}$ $\displaystyle=\int\limits_{0}^{\infty}{\left\{{\exp\left({{-\alpha_{1}y}/m}% \right)\sum_{i=0}^{n\lambda_{1}-1}{\left({{\alpha_{1}}/m}\right)^{i}\frac{y^{i% }}{i!}}}\right\}}\frac{\alpha_{2}^{\lambda_{2}}}{\Gamma\left({\lambda_{2}}% \right)}y^{\lambda_{2}-1}\exp\left({-\alpha_{2}y}\right)dy$ $\displaystyle=\sum_{i=0}^{n\lambda_{1}-1}{\frac{\alpha_{1}^{i}\alpha_{2}^{% \lambda_{2}}}{m^{i}i!\Gamma(\lambda_{2})}\frac{\Gamma(\lambda_{2}+i)}{\left({{% \alpha_{1}}/m+\alpha_{2}}\right)^{\lambda_{2}+i}}}\int\limits_{0}^{\infty}{% \frac{\left({{\alpha_{1}}/m+\alpha_{2}}\right)^{\lambda_{2}+i}}{\Gamma(\lambda% _{2}+i)}}y^{\lambda_{2}+i-1}e^{-\left({{\alpha_{1}}/m+\alpha_{2}}\right)y}dy$ $\displaystyle=\sum_{i=0}^{n\lambda_{1}-1}\frac{\Gamma(\lambda_{2}+i)}{m^{i}i!% \Gamma(\lambda_{2})}\left({\frac{m}{1+\rho}}\right)^{i}\left({\frac{\rho}{1+% \rho}}\right)^{\lambda_{2}}$ (4)

where, $\lambda_{1}$ is considered to be integer and ${\rho=m\alpha_{2}}/{\alpha_{1}}$ . The augmented strength reliability expressions for case-I, case-II and case-III under ASP can be obtained directly by substituting $(k=n=1)$ , $(k=2,m=1)$ and $(k=3)$ respectively in Eq. 4.

3. Maximum likelihood estimation of augmented strength reliability models

This section deals with the maximum likelihood estimation of parameters of augmented strength reliability $\left({R_{k}}\right)$ . Suppose ${{\bm{Z}}}_{{{\bm{k}}}}=(z_{k1},z_{k2},\ldots,z_{kn_{1}})$ and ${{\bm{Y}}}=(y_{1},y_{2},\ldots,y_{n_{2}})$ be the two independent random samples of sizes $n_{1}$ and $n_{2}$ are drawn from the distributions of augmented strength and the common stress respectively. Then the likelihood function is given by

$\displaystyle L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},\lambda_{2}/z_{k},% y}\right)=\left({\frac{\alpha_{1}}{m}}\right)^{nn_{1}\lambda_{1}}\left({\frac{% 1}{\Gamma\left({n\lambda_{1}}\right)}}\right)^{n_{1}}\frac{\alpha_{2}^{n_{2}% \lambda_{2}}}{\Gamma(\lambda_{2})^{n_{2}}}$ $\displaystyle\quad∼{}\exp\left[{-\left({\frac{\alpha_{1}n_{1}\bar{{z}}_{k}}{m}% +\alpha_{2}n_{2}\bar{{y}}}\right)+\left({n\lambda_{1}-1}\right)s_{1}+\left({% \lambda_{2}-1}\right)s_{2}}\right]$ (5)

where,

$\prod\limits_{i=1}^{n_{1}}z_{ki}^{\left({n\lambda_{1}-1}\right)}\prod\limits_{% j=1}^{n_{2}}y_{j}^{\left({\lambda_{2}-1}\right)}=\exp\left[{\left({n\lambda_{1% }-1}\right)s_{1}+\left({\lambda_{2}-1}\right)s_{2}}\right],s_{1}=\sum_{i=1}^{s% }{\log z_{ki}}\text{∼{}and∼{}}s_{2}=\sum_{i=1}^{t}{\log y_{j}}$

The log-likelihood function is given by

$\displaystyle\log L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},\lambda_{2}/z_% {k},y}\right)=nn_{1}\lambda_{1}\left({\log\alpha_{1}-\log m}\right)+n_{2}% \lambda_{2}\log\alpha_{2}-n_{1}\log\left\{{\Gamma(n\lambda_{1})}\right\}$ $\displaystyle\quad∼{}-n_{2}\log\left\{{\Gamma(\lambda_{2})}\right\}-\frac{% \alpha_{1}n_{1}\bar{{z}}_{k}}{m}-\alpha_{2}n_{2}\bar{{y}}+\left({n\lambda_{1}-% 1}\right)s_{1}+\left({\lambda_{2}-1}\right)s_{2}$ (6)

3.1 When both scale

(\alpha_{1},\alpha_{2})

and shape

(\lambda_{1},\lambda_{2})

parameters are unknown

The MLEs $\hat{{\alpha}}_{1},\hat{{\alpha}}_{2}\hat{{\lambda}}_{1}$ and $\hat{{\lambda}}_{2}$ of $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ can be obtained as the simultaneous solution of the following partial derivative equations with respect to $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ respectively, which are given by

$\displaystyle\frac{\partial\log L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},% \lambda_{2}/z_{k},y}\right)}{\partial\alpha_{1}}=0\Rightarrow\hat{{\alpha}}_{1% }=\frac{mn\hat{{\lambda}}_{1}}{\bar{{z}}_{k}}$ (7) $\displaystyle\frac{\partial\log L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},% \lambda_{2}/z_{k},y}\right)}{\partial\alpha_{2}}=0\Rightarrow\hat{{\alpha}}_{2% }=\frac{\hat{{\lambda}}_{2}}{\bar{{y}}}$ (8) $\displaystyle\frac{\partial\log L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},% \lambda_{2}/z_{k},y}\right)}{\partial\lambda_{1}}=nn_{1}\log\left({{\alpha_{1}% }/m}\right)-n_{1}\psi\left({n\lambda_{1}}\right)+ns_{1}=0$ (9) $\displaystyle\frac{\partial\log L_{k}\left({\alpha_{1},\lambda_{1},\alpha_{2},% \lambda_{2}/z_{k},y}\right)}{\partial\lambda_{2}}=n_{2}\log\alpha_{2}-n_{2}% \psi\left({\lambda_{2}}\right)+s_{2}=0$ (10)

where, $\psi\left(.\right)$ is the digamma function, which is defined as $\psi\left(x\right)={\partial\ln\Gamma\left(x\right)}/{\partial x},\vee x>0$ .

It may be noticed from the above likelihood equations, simultaneous trivial solutions are not possible. Thus, the maximum likelihood estimators $\hat{{\alpha}}_{1},\hat{{\alpha}}_{2},\hat{{\lambda}}_{1}$ and $\hat{{\lambda}}_{2}$ can be obtained by using any numerical iterative technique. We therefore propose the Newton-Raphson iterative procedure to find out the simultaneous solution of the above likelihood equations. The ML estimator of $R_{k}$ , denoted as $\hat{{R}}_{k}$ can be easily obtained due to invariance property of MLE by replacing $\lambda_{1}=\hat{{\lambda}}_{1},\lambda_{2}=\hat{{\lambda}}_{2}$ and $\rho=\hat{{\rho}}$ ; where $\hat{{\rho}}={m\hat{{\alpha}}_{2}}/{\hat{{\alpha}}_{1}}$ in the expression of $R_{k}$ . The ML estimates of $R_{k}$ for known values of shape and scale parameters are also discussed as special cases in the following remarks.

Remark 1. when both the shape parameters $(\lambda_{1},\lambda_{2})$ are known then the MLEs $\hat{{\alpha}}_{1}$ and $\hat{{\alpha}}_{2}$ of scale parameters $\alpha_{1}$ and $\alpha_{2}$ can easily be found from Eqs 7 and 8 by keeping fixed values of $\lambda_{1}$ and $\lambda_{2}.$

Remark 2. when the scale parameters $(\alpha_{1},\alpha_{2})$ are known then the MLEs $\hat{{\lambda}}_{1}$ and $\hat{{\lambda}}_{2}$ of shape parameters $\lambda_{1}$ and $\lambda_{2}$ can also be obtained from Eqs 9 and 10 by keeping fixed values of $\alpha_{1}$ and $\alpha_{2}$ .

4. Bayesian estimation of generalized augmenting strength reliability models

In this section, we propose Bayes estimation of $R_{k}(k=1,2,3)$ for generalized case of ASP by assuming the model parameters $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ as independent random variables with non-informative types of priors. In this study, for a better comprehensive comparison of the proposed Bayes estimators of augmented strength reliability, we consider non-informative (uniform and Jeffreys) types of priors under two different loss functions known as symmetric (SELF) and asymmetric (LLF). It is known fact that the squared error loss function considers the overestimation and underestimation as equally penalized, whereas, in LINEX loss function the overestimation is more serious than the underestimation or vice-versa.

4.1 Uniform prior

In this subsection, we assume $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ as independent random variables having prior distribution as uniformly distributed. To find out the Bayes estimators, the following assumption for shape $(\lambda_{1},\lambda_{2})$ and scale $(\alpha_{1},\alpha_{2})$ parameters is considered.

4.1.1 When both scale $(\alpha_{1},\alpha_{2})$ and shape $(\lambda_{1},\lambda_{2})$ parameters are unknown

The joint prior density of random variables $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ is given as follows

$p_{1}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2})\propto\frac{1}{\alpha_{1}% \alpha_{2}\lambda_{1}\lambda_{2}};0<\alpha_{1},\alpha_{2},\lambda_{1},\lambda_% {2}<\infty$ (11)

Thus, the joint posterior probability density is obtained by combining both likelihood function $L_{k}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/$ $z_{k},y)$ and joint prior probability density function $p_{1}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2})$ is given by

$\displaystyle\Pi_{k1}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k% },y}\right)\propto\left\{{\alpha_{1}^{nn_{1}\lambda_{1}-1}e^{\left({\frac{-% \alpha_{1}n_{1}\bar{{z}}_{k}}{m}}\right)}}\right\}\times\left\{{\alpha_{2}^{n_% {2}\lambda_{2}-1}e^{\left({-n_{2}\bar{{y}}\alpha_{2}}\right)}}\right\}$ $\displaystyle\quad∼{}\times\left\{{\frac{1}{\lambda_{1}m^{nn_{1}\lambda_{1}}% \left\{{\Gamma\left({n\lambda_{1}}\right)}\right\}^{n_{1}}}e^{\left({ns_{1}% \lambda_{1}}\right)}}\right\}\times\left\{{\frac{1}{\lambda_{2}\left[{\Gamma% \left({\lambda_{2}}\right)}\right]^{n_{2}}}e^{\left({s_{2}\lambda_{2}}\right)}% }\right\}$ $\displaystyle\propto G_{\alpha_{1}}\left[{\frac{n_{1}\bar{{z}}_{k}}{m},nn_{1}% \lambda_{1}}\right]\times G_{\alpha_{2}}\left[{n_{2}\bar{{y}},n_{2}\lambda_{2}% }\right]$ $\displaystyle\quad∼{}\times\frac{\Gamma(nn_{1}\lambda_{1})e^{\lambda_{1}ns_{1}% }}{\left({{\displaystyle\frac{n_{1}\bar{{z}}_{k}}{m}}}\right)^{(nn_{1}\lambda_% {1})}\lambda_{1}m^{nn_{1}\lambda_{1}}\left\{{\Gamma\left({n\lambda_{1}}\right)% }\right\}^{n_{1}}}\times\frac{\Gamma(n_{2}\lambda_{2})e^{s_{2}\lambda_{2}}}{% \left({n_{2}\bar{{y}}}\right)^{n_{2}\lambda_{2}}\lambda_{2}\left[{\Gamma\left(% {\lambda_{2}}\right)}\right]^{n_{2}}}$ (12)

The marginal posterior densities of $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ are respectively given as

$\displaystyle\pi_{k11}(\alpha_{1}/\lambda_{1},z_{k})\propto G\left({\frac{n_{1% }\bar{{z}}_{k}}{m},nn_{1}\lambda_{1}}\right)$ (13) $\displaystyle\pi_{k12}(\alpha_{2}/\lambda_{2},y)\propto G\left({n_{2}\bar{{y}}% ,n_{2}\lambda_{2}}\right)$ (14) $\displaystyle\pi_{k13}(\lambda_{1}/z_{k})\propto{\displaystyle\frac{\Gamma(nn_% {1}\lambda_{1})e^{\lambda_{1}ns_{1}}}{\left({{\displaystyle\frac{n_{1}\bar{{z}% }_{k}}{m}}}\right)^{(nn_{1}\lambda_{1})}\lambda_{1}m^{nn_{1}\lambda_{1}}\left% \{{\Gamma\left({n\lambda_{1}}\right)}\right\}^{n_{1}}}}$ (15) $\displaystyle\pi_{k14}(\lambda_{2}/y)\propto{\displaystyle\frac{\Gamma(n_{2}% \lambda_{2})e^{s_{2}\lambda_{2}}}{\left({n_{2}\bar{{y}}}\right)^{n_{2}\lambda_% {2}}\lambda_{2}\left[{\Gamma\left({\lambda_{2}}\right)}\right]^{n_{2}}}}$ (16)

It should be noted that the forms of the marginal posterior densities, given in Eqs 15 and 16 are not in any standard distributional forms. In such situations, the Metropolis-Hasting (M-H) algorithm can be recommended to draw random samples from any arbitrary posterior. A detailed discussion on M-H algorithm is given in Section 6. To generate a posterior random sample of size $N$ (say) from the above marginal posterior distributions, we follow the following steps:

Step 1: Step 1:

Start with $q=1$ .

Step 2:

Choose initial value $\lambda_{1}^{(0)},\lambda_{2}^{(0)}$ .

Step 3:

Using M-H algorithm, generate $\lambda_{1:q}\sim\pi_{k13}(./z_{k})$ .

Step 4:

Using M-H algorithm, generate $\lambda_{2:q}\sim\pi_{k14}(./y)$ .

Step 5:

Generate $\alpha_{1:q}\sim\pi_{k11}(./z_{k})$ .

Step 6:

Generate $\alpha_{2:q}\sim\pi_{k12}(./y)$ .

Step 7:

Repeat steps 3 to 6 for $q=1,2,\ldots,N$ .

Under the squared error loss function, the Bayes estimators of augmented strength reliability $\left({R_{k};k=1,2,3}\right)$ is defined by its posterior expectation and is given by

$_{U}\hat{{R}}_{k}^{\textit{self}}=E\left({R_{k}/z_{k},y}\right)=\int\limits_{0% }^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{% \infty}{R_{k}\Pi_{k1}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k% },y}\right)d\alpha_{1}d\alpha_{2}d\lambda_{1}d\lambda_{2}}}}}$ (17)

Under LINEX loss function, the Bayes estimate ( $\hat{{R}}_{k}^{\textit{llf}}$ ) of augmented strength reliability model defined by

$_{U}\hat{{R}}_{k}^{\textit{llf}}=\frac{-1}{p}\ln\left[{E\left({e^{-pR_{k}}/z_{% k},y}\right)}\right]=\frac{-1}{p}\ln\left[{\int\limits_{0}^{\infty}{\int% \limits_{0}^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{\infty}{e^{-pR_% {k}}\Pi_{k1}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k},y}% \right)d\alpha_{1}d\alpha_{2}d\lambda_{1}d\lambda_{2}}}}}}\right]$ (18)

where, $p$ is the scale constant. Thus, the Bayes estimate of $R_{k}$ under SELF and LLF can be approximated through M-H algorithm and are given by

$\displaystyle{}_{U}\hat{{R}}_{k}^{\textit{self}}=\frac{1}{N}\sum_{q=1}^{N}{% \left({R_{k}}\right)_{\alpha_{1}=\alpha_{1:q},\alpha_{2}=\alpha_{2:q},\lambda{% }_{1}=\lambda_{1:q},\lambda_{2}=\lambda_{2:q}}}$ (19) $\displaystyle{}_{U}\hat{{R}}_{k}^{\textit{llf}}=\frac{-1}{p}\ln\left[{e^{-p% \frac{1}{N}\sum_{q=1}^{N}{\left({R_{k}}\right)_{\alpha_{1}=\alpha_{1:q},\alpha% _{2}=\alpha_{2:q},\lambda_{1}=\lambda_{1:q},\lambda_{2}=\lambda_{2:q}}}}}\right]$ (20)

where, $\alpha_{1:q},\alpha_{2:q},\lambda_{1:q}$ and $\lambda_{2:q};q=1,2,\ldots,N$ are the four independent random samples drawn fromEqs 13–16 respectively through M-H algorithm. The Bayes estimates of $R_{k}$ for two special cases of known values of shape and scale parameters are discussed in following remarks.

Remark 3. When the shape parameters $(\lambda_{1},\lambda_{2})$ are assumed to be known then the marginal posteriors $\pi_{k11}(\alpha_{1}/\lambda_{1},$ $z_{k})$ and $\pi_{k12}(\alpha_{2}/\lambda_{2},y)$ , given in Eqs 13 and 14 have closed form as gamma distribution. Thus, in this situation we use importance sampling procedure to draw random posterior samples $\alpha_{1:q}$ and $\alpha_{2:q};q=1,2,\ldots,N$ for fixed values of $\lambda_{1}$ and $\lambda_{2}$ . The approximate Bayes estimates of $R_{k}$ under SELF and LLF can be obtained by substituting $\alpha_{1:q}$ and $\alpha_{2:q}$ in place of $\alpha_{1}$ and $\alpha_{2}$ with fixed values of $\lambda_{1}$ and $\lambda_{2}$ in Eqs 19 and 20 respectively.

Remark 4. When the scale parameters $(\alpha_{1},\alpha_{2})$ are assumed to be known then the marginal posteriors $\pi_{k13}(\lambda_{1}/\alpha_{1},$ $z_{k})$ and $\pi_{k14}(\lambda_{2}/\alpha_{2},y)$ , given in Eqs 15 and 16 are not in any standard distributional form and thus in this situation we propose to use M-H algorithm to draw $N$ random posterior samples $\lambda_{1:q}$ and $\lambda_{2:q};q=1,2,\ldots,N$ for fixed values of $\alpha_{1}$ and $\alpha_{2}.$ The Bayes estimates of $R_{k}$ under SELF and LLF can be approximated by substituting $\lambda_{1:q}$ and $\lambda_{2:q}$ in place of $\lambda_{1}$ and $\lambda_{2}$ with fixed values of $\alpha_{1}$ and $\alpha_{2}$ in Eqs 19 and 20 respectively.

4.2 Jeffreys prior

In this subsection, we assume $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ as independent random variables having non-informative Jeffreys prior. Jeffrey (1998) proposed a type of non-informative prior, which is defined as

$J\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)\propto\sqrt{\det I% \left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)}$ (21)

where, $I\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)$ is Fisher information matrix, defined as follows

$I\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)=-E\begin{bmatrix% }{{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{1}^{2}}}}&{{% \displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{1}\partial\alpha_{2}}}}% &{{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{1}\partial\lambda_{1% }}}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{1}\partial% \lambda_{2}}}}\\ {{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{2}\partial\alpha_{1}}% }}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{2}^{2}}}}&{{% \displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{2}\partial\lambda_{1}}}% }&{{\displaystyle\frac{\partial^{2}\log L}{\partial\alpha_{2}\partial\lambda_{% 2}}}}\\ {{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{1}\partial\alpha_{1}% }}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{1}\partial\alpha% _{2}}}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{1}^{2}}}}&{{% \displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{1}\partial\lambda_{2}}% }}\\ {{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{2}\partial\alpha_{1}% }}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{2}\partial\alpha% _{2}}}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{2}\partial% \lambda_{1}}}}&{{\displaystyle\frac{\partial^{2}\log L}{\partial\lambda_{2}^{2% }}}}\\ \end{bmatrix}$ (22)

To find out the Bayes estimators under Jeffreys prior, the following possible assumptions for shape $(\lambda_{1},\lambda_{2})$ and scale $(\alpha_{1},\alpha_{2})$ parameters are considered.

4.2.1 When both scale

(\alpha_{1},\alpha_{2})

and shape

(\lambda_{1},\lambda_{2})

parameters are unknown

The Fisher information matrix is given as

$I\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)=\begin{bmatrix}{% {\displaystyle\frac{nn_{1}\lambda_{1}}{\alpha_{1}^{2}}}}&0&{-{\displaystyle% \frac{nn_{1}}{\alpha_{1}}}}&0\\ 0&{{\displaystyle\frac{n\lambda_{2}}{\alpha_{2}^{2}}}}&0&{-{\displaystyle\frac% {n_{2}}{\alpha_{2}}}}\\ {-{\displaystyle\frac{nn_{1}}{\alpha_{1}}}}&0&{n_{1}\psi^{\prime}(n\lambda_{1}% )}&0\\ 0&{-{\displaystyle\frac{n_{2}}{\alpha_{2}}}}&0&{n_{2}\psi^{\prime}(\lambda_{2}% )}\\ \end{bmatrix}$ (23)

where, $\psi^{\prime}(.)$ is tri-gamma function, defined as $\psi^{\prime}\left(x\right)={\partial^{2}\ln\Gamma\left(x\right)}/{\partial x^% {2}},\vee x>0$ . Then, the Jeffreys prior of $\alpha_{1},$ $\alpha_{2},$ $\lambda_{1}$ and $\lambda_{2}$ is given as

$J_{1}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}}\right)\propto(% \alpha_{1}\alpha_{2})^{-1}\left[{\lambda_{1}\lambda_{2}\alpha_{2}^{2}\psi^{% \prime}(n\lambda_{1})\psi^{\prime}(\lambda_{2})-\lambda_{1}\psi^{\prime}(n% \lambda_{1})-n\lambda_{2}\psi^{\prime}(\lambda_{2})+n}\right]^{1/2}$ (24)

The joint posterior density $\Pi_{k2}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k},y)$ can be defined by combining both likelihood function $L_{k}(\alpha_{1},$ $\alpha_{2},$ $\lambda_{1},$ $\lambda_{2}/z_{k},y)$ and joint prior probability density function $J_{1}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2})$ and is given by

$\displaystyle\Pi_{k2}(\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k},y)% \propto\left\{{\frac{\alpha_{1}^{nn_{1}\lambda_{1}-1}\alpha_{2}^{n_{2}\lambda_% {2}-1}e^{\left({ns_{1}\lambda_{1}+s_{2}\lambda_{2}}\right)}}{m^{nn_{1}\lambda_% {1}}\Gamma\left({n\lambda_{1}}\right)^{n_{1}}\Gamma(\lambda_{2})^{n_{2}}}\exp% \left[{-\left\{{\frac{\alpha_{1}n_{1}\bar{{z}}_{k}}{m}+\alpha_{2}n_{2}\bar{{y}% }}\right\}}\right]}\right\}$ $\displaystyle\quad∼{}\times\left[{\lambda_{1}\lambda_{2}\alpha_{2}^{2}\psi^{% \prime}(n\lambda_{1})\psi^{\prime}(\lambda_{2})-\lambda_{1}\psi^{\prime}(n% \lambda_{1})-n\lambda_{2}\psi^{\prime}(\lambda_{2})+n}\right]^{1/2}$ (25)

To generate a posterior random sample from the above posterior distribution we follow M-H algorithm discussed in Section 4.1.1.

Under the squared error loss function, the Bayes estimators of augmented strength reliability $\left({R_{k};k=1,2,3}\right)$ is defined by its posterior expectation and is given by

$_{J}\hat{{R}}_{k}^{\textit{self}}=E\left({R_{k}/z_{k},y}\right)=\int\limits_{0% }^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{% \infty}{R_{k}\Pi_{k2}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k% },y}\right)d\alpha_{1}d\alpha_{2}d\lambda_{1}d\lambda_{2}}}}}$ (26)

Under LINEX loss function, the Bayes estimate ( $\hat{{R}}_{k}^{\textit{llf}}$ ) of augmented strength reliability model defined by

$_{J}\hat{{R}}_{k}^{\textit{llf}}=\frac{-1}{p}\ln\left[{E\left({e^{-pR_{k}}/z_{% k},y}\right)}\right]=\frac{-1}{p}\ln\left[{\int\limits_{0}^{\infty}{\int% \limits_{0}^{\infty}{\int\limits_{0}^{\infty}{\int\limits_{0}^{\infty}{e^{-pR_% {k}}\Pi_{k2}\left({\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2}/z_{k},y}% \right)d\alpha_{1}d\alpha_{2}d\lambda_{1}d\lambda_{2}}}}}}\right]$ (27)

where, $p$ is the scale constant. Thus, the Bayes estimate of $R_{k}$ under SELF and LLF can be approximated through M-H algorithm and are given by

$\displaystyle{}_{J}\hat{{R}}_{k}^{\textit{self}}=\frac{1}{N}\sum_{q=1}^{N}{% \left({R_{k}}\right)_{\alpha_{1}=\alpha_{1:q},\alpha_{2}=\alpha_{2:q},\lambda_% {1}=\lambda_{1:q},\lambda_{2}=\lambda_{2:q}}}$ (28) $\displaystyle{}_{J}\hat{{R}}_{k}^{\textit{llf}}=\frac{-1}{p}\ln\left[{e^{-p% \frac{1}{N}\sum_{q=1}^{N}{\left({R_{k}}\right)_{\alpha_{1}=\alpha_{1:q},\alpha% _{2}=\alpha_{2:q},\lambda_{1}=\lambda_{1:q},\lambda_{2}=\lambda_{2:q}}}}}\right]$ (29)

where, $\alpha_{1:q},\alpha_{2:q},\lambda_{1:q}$ and $\lambda_{2:q};q=1,2,\ldots,N$ are the four independent random samples drawn from Eq. 25 respectively through M-H algorithm. The following remarks are the special cases, where, the Bayes estimates of $R_{k}$ for known values of shape and scale parameters have been discussed.

Remark 5. When the shape parameters $(\lambda_{1},\lambda_{2})$ are assumed to be known then the Jeffreys prior of $\alpha_{1}$ and $\alpha_{2}$ becomes $J_{2}(\alpha_{1},\alpha_{2}/\lambda_{1},\lambda_{2})$ $\propto$ $\alpha_{1}^{-1}\alpha_{2}^{-1}$ with their marginal posterior density function as $\pi_{k11}(\alpha_{1}/\lambda_{1},z_{k})\propto G(n_{1}\bar{{z}}_{k}/m,$ $nn_{1}\lambda_{1})$ and $\pi_{k12}(\alpha_{2}/\lambda_{2},y)\propto G\left({n_{2}\bar{{y}},n_{2}\lambda% _{2}}\right)$ respectively. Thus, the Bayes estimates of $R_{k}$ under SELF and LLF can be approximated by substituting $\alpha_{1:q}$ and $\alpha_{2:q}$ in place of $\alpha_{1}$ and $\alpha_{2}$ with fixed values of $\lambda_{1}$ and $\lambda_{2}$ in Eqs 28 and 29 respectively. Where, $\alpha_{1:q}$ and $\alpha_{2:q}$ ; $q=1,2,\ldots,N$ are the generated posterior samples through importance sampling technique.

Remark 6. When the scale parameters $(\alpha_{1},\alpha_{2})$ are assumed to be known then the Jeffreys prior of $\lambda_{1}$ and $\lambda_{2}$ is given as $J_{3}(\lambda_{1},\lambda_{2}/\alpha_{1},\alpha_{2})\propto\left[{\psi^{\prime% }(n\lambda_{1})\psi^{\prime}(\lambda_{2})}\right]^{1/2}$ , then their marginal posteriors $\pi_{k21}(\lambda_{1}/\alpha_{1},z_{k})$ and $\pi_{k22}(\lambda_{2}/\alpha_{2},y)$ do not have any standard distributional forms. Thus, in this situation we propose to use M-H algorithm to generate posterior random samples from any arbitrary density. The approximated Bayes estimates of $R_{k}$ under SELF and LLF can be obtained by substituting $\lambda_{1:q}$ and $\lambda_{2:q}$ in place of $\lambda_{1}$ and $\lambda_{2}$ with fixed values of $\alpha_{1}$ and $\alpha_{2}$ in Eqs 28 and 29 respectively, where, $\lambda_{1:q}$ and $\lambda_{2:q}$ ; $q=1,2,\ldots,N$ are the generated posterior samples through M-H algorithm.

Table 1

Average estimates, MSE and Absolute bias of the estimator of augmented strength reliability $(R_{k})$ under self and llf for uniform and Jeffreys priors for different $(n_{1},n_{2})$ with variation in Model Parameters

\left({s,t}\right)

MLE

Bayes

Uniform prior

Jeffreys prior

SELF

LLF

SELF

LLF

\left({\alpha_{1}=0.5,\alpha_{2}=1.5,\lambda_{1}=0.75,\lambda_{2}=1.5,p=2.5;n=% 2,m=2}\right)

R_{3}=

0.9123701

(20, 20)

Estimate

MSE

Abs.bias

0.867305

0.023933

0.045065

0.867202

0.002558

0.045168

0.866689

0.002612

0.045681

0.813716

0.009753

0.098654

0.813379

0.009820

0.098991

(20, 40)

Estimate

MSE

Abs.bias

0.890417

0.013356

0.021953

0.869242

0.002503

0.043128

0.868706

0.002558

0.043664

0.811775

0.010137

0.100595

0.811455

0.010202

0.100915

(40, 20)

Estimate

MSE

Abs.bias

0.869175

0.019372

0.043195

0.863775

0.002602

0.048595

0.863516

0.002629

0.048854

0.813066

0.009865

0.099304

0.812889

0.009901

0.099481

(50, 50)

Estimate

MSE

Abs.bias

0.89584

0.007106

0.01653

0.861425

0.002824

0.050945

0.861188

0.002851

0.051183

0.816775

0.009140

0.095596

0.816629

0.009167

0.095741

\left({\alpha_{1}=1.5,\alpha_{2}=2.75,\lambda_{1}=1.75,\lambda_{2}=2.5,p=2.5;n% =2,m=2}\right)

R_{3}=

0.9730883

(20, 20)

Estimate

MSE

Abs.bias

0.905227

0.030279

0.067862

0.949731

0.000729

0.023357

0.949507

0.000745

0.023582

0.930253

0.001893

0.042835

0.930188

0.001899

0.0429

(20, 40)

Estimate

MSE

Abs.bias

0.936048

0.012474

0.037040

0.955109

0.000331

0.017979

0.954726

0.000345

0.018363

0.924826

0.002392

0.048262

0.924768

0.002397

0.048321

(40, 20)

Estimate

MSE

Abs.bias

0.902813

0.031989

0.070275

0.946105

0.000736

0.026983

0.945862

0.000749

0.027226

0.931897

0.001731

0.041191

0.931867

0.001734

0.041222

(50, 50)

Estimate

MSE

Abs.bias

0.943144

0.00592

0.029944

0.945346

0.000686

0.027742

0.945159

0.000695

0.02793

0.930727

0.001608

0.042361

0.930678

0.001612

0.042410

\left({\alpha_{1}=\alpha_{2}=1.5,\lambda_{1}=\lambda_{2}=2.5,p=2.5;n=2,m=2}\right)

R_{3}=

0.9703247

(20, 20)

Estimate

MSE

Abs.bias

0.879371

0.044513

0.090954

0.975951

0.000033

0.005626

0.975807

0.000031

0.005482

0.928023

0.001793

0.042302

0.92793

0.001801

0.042395

(20, 40)

Estimate

MSE

Abs.bias

0.917889

0.021901

0.052436

0.976367

0.000039

0.006043

0.976227

0.000038

0.005903

0.927096

0.001871

0.043228

0.927006

0.001879

0.043319

(40, 20)

Estimate

MSE

Abs.bias

0.889315

0.035261

0.081009

0.973485

0.000011

0.003161

0.973407

0.000011

0.003082

0.929189

0.0017

0.041136

0.929139

0.001704

0.041186

(50, 50)

Estimate

MSE

Abs.bias

0.92923

0.011574

0.041095

0.97247

0.000005

0.002145

0.972404

0.000005

0.00208

0.931498

0.001512

0.038826

0.931461

0.001515

0.038864

Table 2

\left({s,t}\right)

MLE

Bayes

Uniform prior

Jeffreys prior

SELF

LLF

SELF

LLF

\left({\alpha_{1}=\alpha_{2}=1.5,\lambda_{1}=\lambda_{2}=2.5,p=2.5;n=2,m=1.5}\right)

R_{3}=

0.9344424

(20, 20)

Estimate

MSE

Abs.bias

0.835698

0.05996

0.098744

0.946149

0.000144

0.011706

0.945833

0.000137

0.011391

0.877274

0.003272

0.057168

0.877164

0.003284

0.057278

(20, 40)

Estimate

MSE

Abs.bias

0.864121

0.0384

0.070321

0.942662

0.000073

0.008219

0.942319

0.000068

0.007877

0.880016

0.002965

0.054427

0.879902

0.002978

0.05454

(40, 20)

Estimate

MSE

Abs.bias

0.823151

0.063121

0.111292

0.938972

0.000021

0.00453

0.938793

0.00002

0.00435

0.881735

0.00278

0.052707

0.881675

0.002787

0.052768

(50, 50)

Estimate

MSE

Abs.bias

0.882894

0.021576

0.051549

0.939002

0.000022

0.00456

0.93886

0.000021

0.004418

0.881219

0.002834

0.053223

0.881172

0.002839

0.05327

\left({\alpha_{1}=\alpha_{2}=1.5,\lambda_{1}=\lambda_{2}=2.5,p=2.5;n=2,m=3.5}\right)

R_{3}=

0.9953218

(20, 20)

Estimate

MSE

Abs.bias

0.950953

0.017208

0.044369

0.996687

0.000002

0.001365

0.996676

0.000002

0.001354

0.974978

0.000416

0.020344

0.974944

0.000418

0.020377

(20, 40)

Estimate

MSE

Abs.bias

0.972226

0.004655

0.023095

0.996704

0.000002

0.001382

0.996692

0.000002

0.00137

0.974917

0.000425

0.020405

0.974884

0.000426

0.020438

(40, 20)

Estimate

MSE

Abs.bias

0.957699

0.013127

0.037623

0.995891

0.000002

0.000569

0.995884

0.000001

0.000562

0.97728

0.000326

0.018042

0.977264

0.000326

0.018058

(50, 50)

Estimate

MSE

Abs.bias

0.97938

0.002601

0.015942

0.995862

0.000001

0.00054

0.995857

0.000001

0.000535

0.978037

0.000299

0.017284

0.978025

0.0003

0.017297

Table 3

\left({s,t}\right)

MLE

Bayes

Uniform prior

Jeffreys prior

SELF

LLF

SELF

LLF

\left({\alpha_{1}=\alpha_{2}=1.5,\lambda_{1}=\lambda_{2}=2.5,p=2.5;n=3,m=2}\right)

R_{3}=

0.996952

(20, 20)

Estimate

MSE

Abs.bias

0.947869

0.019329

0.049083

0.997435

0.000021

0.000483

0.997416

0.000023

0.000464

0.981504

0.000239

0.015448

0.98149

0.00024

0.015462

(20, 40)

Estimate

MSE

Abs.bias

0.967762

0.006988

0.02919

0.997311

0.000003

0.000359

0.997291

0.000002

0.000339

0.981837

0.000229

0.015115

0.981824

0.00023

0.015128

(40, 20)

Estimate

MSE

Abs.bias

0.946784

0.020218

0.050168

0.997020

0.000024

0.000068

0.997012

0.000045

0.000060

0.983677

0.000177

0.013275

0.983671

0.000177

0.013281

(50, 50)

Estimate

MSE

Abs.bias

0.979778

0.002412

0.017174

0.997066

0.000042

0.000114

0.997061

0.000052

0.000109

0.98333

0.000186

0.013622

0.983325

0.000186

0.013627

\left({\alpha_{1}=\alpha_{2}=1.5,\lambda_{1}=\lambda_{2}=2.5,p=2.5;n=5,m=2}\right)

R_{3}=

0.999976

(20, 20)

Estimate

MSE

Abs.bias

0.983765

0.005483

0.016211

0.999686

0.000012

0.000290

0.999671

0.000002

0.000305

0.999066

0.000001

0.00091

0.999066

0.000001

0.00091

(20, 40)

Estimate

MSE

Abs.bias

0.995475

0.000387

0.004502

0.999718

0.000003

0.000258

0.999704

0.000029

0.000272

0.999064

0.000001

0.000912

0.999064

0.000001

0.000912

(40, 20)

Estimate

MSE

Abs.bias

0.988018

0.003472

0.011958

0.999921

0.000001

0.000056

0.999920

0.000001

0.000056

0.99914

0.000001

0.000836

0.99914

0.000001

0.000836

(50, 50)

Estimate

MSE

Abs.bias

0.998528

0.000034

0.001448

0.999938

0.000002

0.000038

0.999937

0.000002

0.000039

0.999172

0.000001

0.000804

0.999172

0.000001

0.000804

Table 4

Fitting summary for Carbon fiber breaking strength data for example 2

Data	Model	K-S	LogL	AIC	BIC
X	Gamma	0.0879	$-$ 53.01	110.0204	114.4886
Y		0.0872	$-$ 56.75	117.4958	121.7820
X	Log-normal	0.1173	$-$ 58.21	120.4179	124.8861
Y		0.1059	$-$ 60.98	125.9492	130.2355
X	Exponential	0.3632	$-$ 95.08	192.1612	194.3953
Y		0.2746	$-$ 77.51	157.0216	159.1648

5. Data analysis

This section presents the illustration of the proposed ML and Bayesian estimators of augmented strength reliability for the generalized case of ASP through simulated as well as real data sets.

Example 1. Here we consider the simulated data analysis of the proposed augmented strength reliability. Two samples of sizes 40 and 30 are drawn from the distributions of augmented strength and the common stress i.e. from $gamma\left({{\alpha_{1}}/m,n\lambda_{1}}\right)$ and $gamma\left({\alpha_{2},\lambda_{2}}\right)$ respectively by fixing the parameters $\alpha_{1}=\alpha_{2}=1.50,\lambda_{1}=\lambda_{2}=2.50$ and $m=n=2$ . The calculated true value of augmented strength reliability for case-III is given as $R_{3}=$ 0.970325 with its ML estimate $\hat{{R}}_{3}=$ 0.998448. The Bayes estimates for uniform and Jeffrey’s priors under squared error and LINEX loss are computed by generating fifteen thousand samples through Metropolis-Hasting algorithm. Thus the Bayes estimate of augmented strength reliability for uniform and Jeffreys priors under SELF (LLF) are 0.972275 (0.972189) and 0.953223 (0.954163) respectively.

Example 2. For Real Data Sets (Carbon fiber breaking strength): In this example we tried to fit and analyze the real strength data sets reported in Badar and Priest (1982) using the two parameters gamma distribution. Many authors have used these data sets to illustrate different sets of distributions; e.g. Raqab et al. (2008) and Ali et al. (2012) have carried out for 3-parameter generalized exponential distribution and 4-parameter generalized gamma distribution respectively and they observed that the data sets fit quite well for the underlying distributions. Here we consider two parameters gamma distribution to analyze the following data sets. We, therefore, tested the goodness of fit of the following data for two parameter gamma distribution using Kolmogorov-Smirnov (KS) statistic, Akaike information criterion (AIC) and Bayesian information criterion (BIC). The fitting summary of gamma distribution is given in the following Table 4 which clearly shows that the gamma distribution fits well as compared to log-normal and exponential distributions. The graphs of fitted and empirical distribution functions along with P-P plots for both the data sets $x$ and $y$ for two-parameter gamma, log-normal and exponential distributions are given in Figs 1–3, which clearly show that the underlying datasets give best fit to the two-parameter gamma distribution.

Figure 1.

Fitted and empirical cumulative distribution function for data sets $x$ and $y$ .

Figure 2.

P-P plot for data set $x$ .

Figure 3.

P-P plot for data set $y$ .

The data sets initially reported by Barbar and Priest (1982), represent the strength, measured in GPa, for single carbon fibers and impregnated 1000-carbon fiber tows. Single fibers were tested under tension at gauge lengths of 1, 10, 20, and 50 mm. Impregnated tows of 1000 fibers were tested at gauge lengths of 20, 50, 150 and 300 mm. We analyze here the transformed strength data sets by Raqab and Kundu (2005). The data sets are consider the single fibers of 20 mm (Data Set I) and 10 mm (Data Set II) in gauge length, with sample sizes $n=$ 69 and $m=$ 63, respectively. The data are presented below as:

Data set 1 ( $x$ :)	0.312	0.314	0.479	0.552	0.700	0.803	0.861	0.865	0.944	0.958	0.966
0.997	1.006	1.021	1.027	1.055	1.063	1.098	1.140	1.179	1.224	1.240	1.253
1.270	1.272	1.274	1.301	1.301	1.359	1.382	1.382	1.426	1.434	1.435	1.478
1.490	1.511	1.514	1.535	1.554	1.566	1.57	1.586	1.629	1.633	1.642	1.648
1.684	1.697	1.726	1.770	1.773	1.800	1.809	1.818	1.821	1.848	1.880	1.954
2.012	2.067	2.084	2.090	2.096	2.128	2.233	2.433	2.585	2.585

Data set 2 ( $y$ :)	0.101	0.332	0.403	0.428	0.457	0.55	0.561	0.596	0.597	0.645	0.654
0.674	0.718	0.722	0.725	0.732	0.775	0.814	0.816	0.818	0.824	0.859	0.857
0.938	0.940	1.056	1.117	1.128	1.137	1.137	1.177	1.196	1.230	1.325	1.339
1.345	1.420	1.423	1.435	1.443	1.464	1.472	1.494	1.532	1.546	1.577	1.608
1.635	1.693	1.701	1.737	1.754	1.762	1.828	2.052	2.071	2.086	2.171	2.224
2.227	2.425	2.595	3.220

Assuming data set 1 ( $x$ ) as strength and data set 2 ( $y$ ) as stress, the generalized case (i.e., case-III) of ASP was applied to the strength data to augment the strength of the carbon-fiber. In order to obtain the enhanced strength data sets, we added two units each having strength 0.004 (two times of initial stress (0.002)) to the existing strength of carbon fiber. Thus the true value of augmented strength reliability $\left({R_{3}}\right)$ for generalized case of ASP is given by $R_{3}=$ 0.970325. The maximum likelihood estimates of $R_{3}$ based on augmented strength data set is $\hat{{R}}_{3}=$ 0.992728. The Bayes estimates of augmented strength reliability under square error and LINEX loss functions for uniform and Jeffrey’s priors have been carried out through Metropolis-Hasting algorithm. Fifteen thousand random samples of $\alpha_{1},\alpha_{2},\lambda_{1}$ and $\lambda_{2}$ are drawn from their respective marginal posterior distributions by considering Normal density as candidate density to evaluate the Bayes estimates of augmented strength reliability for different priors. Thus the Bayes estimate of augmented strength reliability under SELF (LLF) for uniform and Jeffreys priors are given by 0.987012 (0.986999) and 0.961924 (0.968371) respectively.

6. Simulation study

In this section, the simulation study has been carried out for different combinations of sample sizes and stress-strength reliability parameters for 1000 replications. We studied the behavior of generalized augmented strength reliability parameters under the proposed augmentation strategy plan through simulated samples. The performance of proposed maximum likelihood estimates and Bayes estimates of generalized augmented strength reliability for two non-informative priors under two different loss functions (i.e. SELF and LLF) were compared through their mean square errors (MSE) and absolute biases.

Uniform and Jeffreys priors are considered under the non-informative prior set up and the Bayes estimates were calculated under SELF and LLF. The expressions of Bayes estimates of augmented strength reliability for both the uniform priors are not in explicit forms and cannot be solved analytically and it is also to be noticed that the marginal posteriors are also not in any standard distributional forms. In such a situation, the well-known MCMC technique viz. Metropolis-Hastings algorithm (Hasting, 1970) can be recommended for drawing the samples from any arbitrary posterior distribution. For generating a random sample of size $N$ (say) from a posterior distribution $\pi\left({{{\bm{\theta}}}/data}\right);{{\bm{\theta}}}=(\alpha_{1},\alpha_{2},% \lambda_{1},\lambda_{2})^{\prime}$ , the basic Metropolis-Hastings algorithm consists the following steps:

Step 1: Step 1:
Choose initial value for the parameter ${{\bm{\theta}}}^{(0)}=({\alpha_{1}^{(0)},\alpha_{2}^{(0)},\lambda_{1}^{(0)},% \lambda_{2}^{(0)}})$ such that $\pi({{{\bm{\theta}}}^{(0)}})>0$ .
Step 2:
For $j=1,2,\ldots,N$ repeat the following steps

iii. i.
Set ${{\bm{\theta}}}={{\bm{\theta}}}^{(j-1)}$ .
ii.
Draw a ‘candidate’ value ${{\bm{\theta}}}^{c}$ from a proposal density say $q({{\bm{\theta}}}^{c}/{{\bm{\theta}}})$ .
iii.
Generate ‘U’ uniform variate on range 0 and 1, $u\sim U(0,1)$ .
iv.
If $u\leqslant\min(1,R)$ , accept the candidate point $\left({{{\bm{\theta}}}^{c}}\right)$ with probability

$\min\left\{{1,R=\frac{\pi\left({{{\bm{\theta}}}^{c}/{\textbf{data}}}\right)q% \left({{{\bm{\theta}}}/{{\bm{\theta}}}^{c}}\right)}{\pi\left({{{\bm{\theta}}}/% {\textbf{data}}}\right)q\left({{{\bm{\theta}}}^{c}/{{\bm{\theta}}}}\right)}}% \right\},$

otherwise set ${{\bm{\theta}}}^{(j)}={{\bm{\theta}}}.$

We assume asymptotic normal distribution as proposal distribution. The initial values $\alpha_{1}^{(0)}=\hat{{\alpha}}_{1},$ $\alpha_{2}^{(0)}=\hat{{\alpha}}_{2}$ and $\lambda_{1}^{(0)}$ $=$ $\hat{{\lambda}}_{1},$ $\lambda_{2}^{(0)}=\hat{{\lambda}}_{2}$ are fixed as the ML estimates along with their asymptotic variance covariance matrix to draw the initial random samples from proposal density. To evaluate the Bayes estimate of augmented strength reliability model the Metropolis-Hastings algorithm was carried out with ten thousands of intermediate iterations. The first thousand samples were discarded as burn-in period of Markov chain. We also tested the autocorrelation and it is noticed that the chains are highly auto correlated. For reducing the autocorrelation within the chain, we thinned the chain equally spaced at every second simulation.

The comparisons among the proposed estimators of augmented strength reliability have been made through mean square errors (MSE) and absolute biases for varying values of stress-strength reliability parameters $\alpha_{1},\alpha_{2},\lambda_{1},\lambda_{2},m$ and $n$ and the effect of choice of sample sizes $\left({n_{1},n_{2}}\right)$ are also taken into account. The following Tables 1–3 contain the average estimates, mean square errors (mse) and absolute biases for MLE and Bayes estimates for uniform and Jeffreys priors under SELF as well as LLF of generalized augmented strength reliability. The following observations are made based on the given tables.

•
In Table 1, variation in the model parameters are presented and it is noticed that the mean square errors and absolute biases of all the estimators are in decreasing pattern as the sample sizes increases. It is also observed from the table that the Bayes estimates of augmented strength reliability models under both the loss functions (SELF and LLF) give better result with minimum MSE and absolute biases as compared to ML estimates.
•
The variation in $m(1.5,3.5)$ is tabulated in Table 2 and it may be observed that the Bayes estimates for both the priors under SELF and LLF dominate over the ML estimators. The Bayes estimator for uniform prior under squared error loss function gives more precise result among all the estimators.
•
Table 3 contains the results for the variation in $n(3,5)$ and it is observed from the table that the MSEs and absolute biases decrease for the increasing values of sample sizes. It is also observed that for higher values of $n$ the Bayes estimators perform far better than the ML estimates.

7. Conclusion

In this article, a comparative study between ML and Bayes methods of estimation are carried out on the basis of their mean square errors and absolute biases. In Bayesian paradigm, two types of non-informative priors (uniform and Jeffreys priors) were taken into account and the Bayes estimates were computed under SELF and LLF. On the basis of performed simulation study, it may be concluded that the Bayes estimators of augmented strength reliability under both SELF and LLF perform better than ML estimators. Even, Bayes estimators for uniform prior under squared error loss function gives more precise result in comparison to Jeffreys priors. It is seen from the table that the proposed augmentation strategy performs well in enhancing the strength of the weaker equipment. The augmented strength reliability gets enhanced by more up to 99% as we increase the augmentation parameters ( $m$ and $n$ ). Thus the proposed ASP is suggestive up to some desired level to overcome the early/frequent failure situations.

Footnotes

Acknowledgments

The authors would like to thank to co-editor-in-chief for providing constructive comments of both the referee’s in improving the contents and quality standard of article and also thanks to the University Grants Commission, Govt. of India, New Delhi, India for providing financial support to carry out the proposed research work under the Major Research Project (ref no. F.42-38/2013 (SR) dated March 12th, 2013).

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On estimation of augmented strength reliability parameters under non-informative priors: A non-identical case

Abstract

Keywords

1. Introduction

4.1 Uniform prior

4.1.1 When both scale ( α 1 , α 2 ) and shape ( λ 1 , λ 2 ) parameters are unknown

Footnotes

Acknowledgments

References

4.1.1 When both scale $(\alpha_{1},\alpha_{2})$ and shape $(\lambda_{1},\lambda_{2})$ parameters are unknown