Bayesian and classical estimation of reliability in a multicomponent stress-strength model under adaptive hybrid progressive censored data

Abstract

The statistical inference of multicomponent stress-strength reliability under the adaptive Type-II hybrid progressive censored samples for the Weibull distribution is considered. It is assumed that both stress and strength are two Weibull independent random variables. We study the problem in three cases. First assuming that the stress and strength have the same shape parameter and different scale parameters, the maximum likelihood estimation (MLE), approximate maximum likelihood estimation (AMLE) and two Bayes approximations, due to the lack of explicit forms, are derived. Also, the asymptotic confidence intervals, two bootstrap confidence intervals and highest posterior density (HPD) credible intervals are obtained. In the second case, when the shape parameter is known, MLE, exact Bayes estimation, uniformly minimum variance unbiased estimator (UMVUE) and different confidence intervals (asymptotic and HPD) are studied. Finally, assuming that the stress and strength have the different shape and scale parameters, ML, AML and Bayesian estimations on multicomponent reliability have been considered. The performances of different methods are compared using the Monte Carlo simulations and for illustrative aims, one data set is investigated.

Appendix A

For three parameters case, we compute 13 at $\widehat{\Theta }=(\widehat{\theta }_1,\widehat{\theta }_2,\widehat{\theta }_3)$, where

$$\begin{aligned} d_i&=\rho _1\sigma _{i1}+\rho _2\sigma _{i2}+\rho _3\sigma _{i3}, ~~i=1,2,3,\\ d_4&=u_{12}\sigma _{12}+u_{13}\sigma _{13}+u_{23}\sigma _{23},\\ d_5&=\frac{1}{2}(u_{11}\sigma _{11}+u_{22}\sigma _{22}+u_{33}\sigma _{33}),\\ A&=\ell _{111}\sigma _{11}+2\ell _{121}\sigma _{12}+2\ell _{131}\sigma _{13} +2\ell _{231}\sigma _{23}+\ell _{221}\sigma _{22} +\ell _{331}\sigma _{33},\\ B&=\ell _{112}\sigma _{11}+2\ell _{122}\sigma _{12}+2\ell _{132}\sigma _{13} +2\ell _{232}\sigma _{23}+\ell _{222}\sigma _{22} +\ell _{332}\sigma _{33},\\ C&=\ell _{113}\sigma _{11}+2\ell _{123}\sigma _{12}+2\ell _{133}\sigma _{13} +2\ell _{233}\sigma _{23}+\ell _{223}\sigma _{22} +\ell _{333}\sigma _{33}. \end{aligned}$$

In our case, for $(\theta _1,\theta _2,\theta _3)\equiv (\alpha ,\theta ,\lambda )$ and $u\equiv u(\alpha ,\theta ,\lambda )=R_{s,k}$ as provided in (3), we have

$$\begin{aligned} \rho _1&=\frac{a_1-1}{\alpha }-b_1,~~ \rho _2=-\frac{a_2+1}{\theta } +\frac{b_2}{\theta ^2},~~~ \rho _3=-\frac{a_3+1}{\lambda }+\frac{b_3}{\lambda ^2},\\ \ell _{11}&=-\frac{n(k+1)}{\alpha ^2}-\frac{1}{\lambda }\bigg [\sum _{i=1}^n y_i^\alpha \log ^2(y_i)+\sum _{i=1}^{J_1}S_iy_i^\alpha \log ^2(y_i) +S_ny_n^\alpha \log ^2(y_n)\bigg ]\\&\quad -\; \frac{1}{\theta }\bigg [\sum _{i=1}^n\sum _{j=1}^k x_{ij}^\alpha \log ^2(x_{ij})+\sum _{i=1}^n\sum _{j=1}^{J_2}R_{ij} x_{ij}^\alpha \log ^2(x_{ij})+\sum _{j=1}^{n}R_{ik} x_{ik}^\alpha \log ^2(x_{ik})\bigg ],\\ \ell _{12}&=\frac{1}{\theta ^2}\bigg [\sum _{i=1}^{n}\sum _{j=1}^k x_{ij}^\alpha \log (x_{ij}) +\sum _{i=1}^{n}\sum _{j=1}^{J_2}R_{ij}x_{ij}^\alpha \log (x_{ij}) +\sum _{i=1}^{n}R_{ik}x_{ik}^\alpha \log (x_{ik})\bigg ]=\ell _{21},\\ \ell _{13}&=\frac{1}{\lambda ^2}\bigg [\sum _{i=1}^{n}y_{i}^\alpha \log (y_i) +\sum _{i=1}^{J_1}S_iy_{i}^\alpha \log (y_i) +S_ny_{n}^\alpha \log (y_n)\bigg ]=\ell _{31},\\ \ell _{22}&=\frac{nk}{\theta ^2}-\frac{2}{\theta ^3}\bigg [\sum _{i=1}^{n} \sum _{j=1}^kx_{ij}^\alpha +\sum _{i=1}^{n}\sum _{j=1}^{J_2}R_{ij}x_{ij}^\alpha +\sum _{i=1}^{n}R_{ik}x_{ik}^\alpha \bigg ],\\ \ell _{23}&=0=\ell _{32},\\ \ell _{33}&=\frac{n}{\lambda ^2}-\frac{2}{\lambda ^3} \bigg [\sum _{i=1}^{n}y_{i}^\alpha +\sum _{i=1}^{J_1}S_iy_{i}^\alpha +S_ny_{n}^\alpha \bigg ]. \end{aligned}$$

Also, by using $\ell _{ij}, i,j = 1,2,3, \sigma _{ij}, i,j = 1,2,3$ can be obtained and

$$\begin{aligned} \ell _{111}&=\frac{2n(k+1)}{\alpha ^3}-\frac{1}{\lambda }\bigg [\sum _{i=1}^n y_i^\alpha \log ^3(y_i)+\sum _{i=1}^{J_1}S_iy_i^\alpha \log ^3(y_i) +S_ny_n^\alpha \log ^3(y_n)\bigg ]\\&-\frac{1}{\theta }\bigg [\sum _{i=1}^n\sum _{j=1}^k x_{ij}^\alpha \log ^3(x_{ij})+\sum _{i=1}^n\sum _{j=1}^{J_2}R_{ij} x_{ij}^\alpha \log ^3(x_{ij})+\sum _{j=1}^{n}R_{ik} x_{ik}^\alpha \log ^3(x_{ik})\bigg ],\\ \ell _{112}&=\frac{1}{\theta ^2}\bigg [\sum _{i=1}^{n}\sum _{j=1}^kx_{ij}^\alpha \log ^2(x_{ij}) +\sum _{i=1}^{n}\sum _{j=1}^{J_2}R_{ij}x_{ij}^\alpha \log ^2(x_{ij}) +\sum _{i=1}^{n}R_{ik}x_{ik}^\alpha \log ^2(x_{ik})\bigg ]\\&=\ell _{211}=\ell _{121},\\ \ell _{113}&=\frac{1}{\lambda ^2}\bigg [\sum _{i=1}^{n}y_{i}^\alpha \log ^2(y_i) +\sum _{i=1}^{J_1}S_iy_{i}^\alpha \log ^2(y_i) +S_ny_{n}^\alpha \log ^2(y_n)\bigg ]\\&=\ell _{311}=\ell _{131},\\ \ell _{122}&=-\frac{2}{\theta ^3}\bigg [\sum _{i=1}^{n}\sum _{j=1}^k x_{ij}^\alpha \log (x_{ij}) +\sum _{i=1}^{n}\sum _{j=1}^{J_2}R_{ij}x_{ij}^\alpha \log (x_{ij}) +\sum _{i=1}^{n}R_{ik}x_{ik}^\alpha \log (x_{ik})\bigg ]\\&=\ell _{221}=\ell _{212},\\ \ell _{133}&=-\frac{2}{\lambda ^3}\bigg [\sum _{i=1}^{n}y_{i}^\alpha \log (y_i) +\sum _{i=1}^{J_1}S_iy_{i}^\alpha \log (y_i) +S_ny_{n}^\alpha \log (y_n)\bigg ]=\ell _{331}=\ell _{313},\\ \ell _{222}&=-\frac{2nk}{\theta ^3}+\frac{6}{\theta ^4}\bigg [\sum _{i=1}^{n} \sum _{j=1}^kx_{ij}^\alpha +\sum _{i=1}^{n}\sum _{j=1}^{J_2}R_{ij}x_{ij}^\alpha +\sum _{i=1}^{n}R_{ik}x_{ik}^\alpha \bigg ],\\ \ell _{333}&=-\frac{2n}{\lambda ^3}+\frac{6}{\lambda ^4}\bigg [\sum _{i=1}^{n}y_{i}^\alpha +\sum _{i=1}^{J_1}S_iy_{i}^\alpha +S_ny_{n}^\alpha \bigg ], \end{aligned}$$

and other $\ell _{ijk}=0$. In addition, $u_1=\partial R_{s,k}/\partial \alpha =0$, $u_{i1}=\partial ^2 R_{s,k}/(\partial \theta _i\partial \alpha )=0$, $i=1,2,3$, and $u_2$, $u_3$ are given in (10) and (11), respectively. Also,

$$\begin{aligned} u_{22}&=\sum _{p=s}^{k}\sum _{q=0}^{k-p}\left( {\begin{array}{c}k\\ p\end{array}}\right) \left( {\begin{array}{c}k-p\\ q\end{array}}\right) \frac{(-1)^{q+1}2\lambda (p+q)}{(\theta +\lambda (p+q))^3},\\ u_{23}&=\sum _{p=s}^{k}\sum _{q=0}^{k-p}\left( {\begin{array}{c}k\\ p\end{array}}\right) \left( {\begin{array}{c}k-p\\ q\end{array}}\right) \frac{(-1)^{q+1}(p+q)(\lambda (p+q)-\theta )}{(\theta +\lambda (p+q))^3}=u_{32},\\ u_{33}&=\sum _{p=s}^{k}\sum _{q=0}^{k-p}\left( {\begin{array}{c}k\\ p\end{array}}\right) \left( {\begin{array}{c}k-p\\ q\end{array}}\right) \frac{(-1)^{q}2\theta (p+q)^2}{(\theta +\lambda (p+q))^3}. \end{aligned}$$

Hence,

$$\begin{aligned} d_4&=u_{23}\sigma _{23},\qquad d_5=\frac{1}{2}(u_{22}\sigma _{22} +u_{33}\sigma _{33}),\\ A&=\ell _{111}\sigma _{11}+2\ell _{121}\sigma _{12} +2\ell _{131}\sigma _{13}+\ell _{221}\sigma _{22}+\ell _{331}\sigma _{33},\\~~~ B&=\ell _{112}\sigma _{11}+2\ell _{122}\sigma _{12}+\ell _{222}\sigma _{22},\quad C=\ell _{113}\sigma _{11}+2\ell _{133}\sigma _{13}+\ell _{333}\sigma _{33}. \end{aligned}$$