1 Introduction

The single-component stress–strength reliability of a system is defined as the probability \(P(X>Y)\); where, X and Y represents the random strength and stress of the system, respectively. It is obvious that the system fails iff the stress enforced to the system transcends the strength of the system. Estimation of reliability in single component set-up has always attracted the authors. In the literary work, considerable work is available on the Bayesian and non-Bayesian estimation procedures for various single-component stress–strength models. For deep insight, researchers may go through [1,2,3,4,5].

When any engineering system or structure are handovered to be used in real life, there are so many factors, that can affect the reliability of the system or structure. In practice, these factors may be external shocks, operating environments, weights, etc. Usually, these factors may be considered as stresses that the system or structure felt during operation. In such cases, reliability of the system or structure is generally measured through multicomponent stress–strength model. Moreover, advance engineering structures are usually prepared with the highest distinction. For instance, a cloud storage server and commercial aircraft usually have some spare parts and engines, respectively, to deal with sudden breakdowns. For more understanding of these models, one can consider the example of a hanging bridge where the deck is sustained by a series of k vertical cables. Moreover, the bridge will only sustain if at least s vertical cables are not damaged or broken when subjected to a common stress. These type of systems are known as s-out-of-k: G systems {see [6]}

If \(X_i,~i=1,2,...,k\) are k identical random strength variables, which are exposed to a common stress Y so that stress–strength variables are independent, then the reliability in multicomponent stress–strength set-up is given by {see [7]}

$$\begin{aligned} R_{s,k}=\displaystyle \sum _{i=s}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) \displaystyle \int _{-\infty }^{\infty } \left[1-F(y)\right] ^{i}\left[F(y)\right] ^{k-i}dG(y),~s\le k, \end{aligned}$$
(1)

where \(G(\cdot)\) represents the continuous distribution function of the stress variable Y and \(F(\cdot)\) represents the common continuous distribution function of strength variables \(X_i,~i=1,2,...,k\). Here \(R_{s,k}\) denotes the reliability in a multicomponent stress–strength set-up.

Various authors have developed the estimation procedures of reliability in a multicomponent stress–strength set-up under different environmental conditions. To cite a few, one may refer to [6, 8,9,10,11,12,13,14,15]. Recently, Ref. [16] developed the estimation procedures for both Bayesian and non-Bayesian reliability in multicomponent stress–strength model for non-identical strength components. They have also shown the applicability of the obtained estimators for wind speed data. Ref. [17] developed reliability estimation procedures when stress and strength are linked by Fralie-Gumble-Morgenstern copula with Rayleigh marginals. For deep insight on applications of copula and two-phase successive sampling under random non-response, one may refer to [18, 19].

Several authors have done intensive work using proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models. For brief review, of these models one may go through [20,21,22].

The present literature on multicomponent stress–strength reliability mainly focusses on models with stress and strength components belonging to the same family, either with the same or different parameters. The objective of this present paper is to further explore these models in a general framework. For estimating the reliability function in a multicomponent stress–strength set-up, we have introduced a new concept in which the stress and strength variables belong to different generalized family of distributions. Maximum likelihood (ML) and Markov Chain Monte Carlo (MCMC) estimates are obtained under the developed framework. Prior distributions play an essential role for derivation of Bayes estimators and in order to obtain the Bayes estimators of \(R_{s,k}\) by using MCMC method, independent gamma priors are considered. [6, 23] have discussed the selection of gamma priors. Moreover, gamma priors are one of the most popular choices of prior selection, especially when the support of the parameter is positive.

The layout of this paper is manyfold. In Sect. 2, the models are discussed in detail. In Sect. 3, ML estimator of the multicomponent reliability function is derived considering strength population as PHR model and stress population as PRHR model. Fisher information matrix and asymptotic confidence interval (ACI) of \(R_{s,k}\) are also constructed. In Sect. 4, under the same set-up, Bayesian method of estimation by using MCMC method is discussed. In Sect. 5, ML estimator of the multicomponent reliability function is derived considering strength population as PRHR model and stress population as PHR model. Fisher information matrix and ACI of \(R_{s,k}\) are also constructed. In Sect. 6, under the same set-up, the Bayesian method of estimation using MCMC method is discussed. Thereafter, in Sect. 7, a simulation study is exercised to compare the various estimates of \(R_{s,k}\). Furthermore, in Sect. 8, real data example is presented. Lastly, discussions and conclusions are provided in Sect. 9.

2 Multicomponent Stress–strength Model

If a random variable (rv) X follows PHR model with shape parameter \(\tau >0\), then the cumulative distribution function (cdf) and the probability density function (pdf) of X are given by

$$\begin{aligned} F(x;\tau)=1-\left[1-H(x)\right] ^{\tau };~x>0, \end{aligned}$$
(2)

and

$$\begin{aligned} f(x;\tau)=\tau h(x)\left[1-H(x)\right] ^{\tau -1};~x>0, \end{aligned}$$
(3)

respectively. Here \(H(\cdot)\) is the baseline cdf, \(h(\cdot)\) denotes the first derivative of \(H(\cdot)\), such that \(H(0)=0\) and \(H(\infty)=1\). The family of distribution given in (2) represents the well known PHR model in lifetime experiments {see [24, 25]}.

Now, a rv X follows a PRHR model with shape parameter \(\tau >0\), if the cdf and pdf of X are given by

$$\begin{aligned} F(x;\tau)=\left[H( x)\right] ^{\tau };~x>0, \end{aligned}$$
(4)

and

$$\begin{aligned} f(x;\tau)=\tau h(x)\left[H(x)\right] ^{\tau -1};~x>0, \end{aligned}$$
(5)

respectively. The family of distribution given in (4) represents the well known PRHR model in lifetime experiments {see [26]}.

In this paper, a rv X following PHR model with power parameter \(\tau\) will be denoted by \(X\sim\) PHR(\(\tau\)) and following PRHR model will be denoted by \(X\sim\) PRHR(\(\tau\)).

3 ML estimation of \(R_{s,k}\), when Strength Population is PHR Model and Stress Population is PRHR Model

This section deals with the ML estimation of \(R_{s,k}\), when the strength variates follow the PHR model and stress variate follow the PRHR model. Also, it is assumed that \(Y, X_i,~i=1,2,...,k\) are independent. Let PHR(\(\tau _{1}\)) denotes the common pdf of the independent strength variates, \(X_i,~i=1,2,...,k\) and PRHR(\(\tau _{2}\)) denotes the pdf of Y. Using (1), (2) and (4), the reliability \(R_{s,k}\) comes out to be

$$\begin{aligned} R_{s,k}&=\displaystyle \sum _{i=s}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) \tau _{2}\int _{0}^{\infty } \left[1-H( y)\right] ^{\tau _{1}i}\left[1-\left[1-H( y)\right] ^{\tau _{1}}\right] ^{k-i} \left[H( y)\right] ^{\tau _{2}-1} h( y)dy\nonumber \\&=\Psi _1 \tau _{2}\int _{0}^{\infty } [1-H( y)]^{\tau _{1} (i+j)}\left[H( y)\right] ^{\tau _{2}-1} h( y)dy, \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text {where}~ \Psi _1=\displaystyle \sum _{i=s}^{k}\sum _{j=0}^{k-i}\left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}k-i\\ j\end{array}}\right) (-1)^{j}\nonumber \\&=\Psi _1\tau _{2}\int _{0}^{1} z^{\tau _{2}-1}\left( 1-z\right) ^{\tau _{1}(i+j)}dz,~\text {where}~z=H( y)\nonumber \\&=\Psi _1\tau _{2} B(\tau _{2},\tau _{1}(i+j)+1), \end{aligned}$$
(6)

where B(a, b)=\(\dfrac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)}\) is the beta function.

Let us assume that n systems are put on life testing experiment from strength and stress populations, respectively and under this set-up the following data is observed: \(X_{ij}\) and \(Y_{i}\), \(i=1,2,...,n\), \(j=1,2,...,k\). Thus, the likelihood and log-likelihood functions corresponding to the above data set are given by

$$\begin{aligned}&L_1(\tau _{1},\tau _{2}|X,\underline{y})=\prod _{i=1}^{n}\left( \prod _{j=1}^{k} f_{X}(x_{ij};\tau _{1})\right) f_{Y}(y_{i};\tau _{2}) \end{aligned}$$
(7)

and

$$\begin{aligned} l_1(\tau _{1},\tau _{2}|X,\underline{y})&=nk\ln \tau _{1}+n\ln \tau _{2}+(\tau _{1}-1) \displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{k} \ln \left[1-H\left( x_{ij}\right) \right] \nonumber \\&+\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{k}\ln h( x_{ij}) +\left( \tau _{2}-1\right) \displaystyle \sum _{i=1}^{n} \ln H( y_{i})+\displaystyle \sum _{i=1}^{n}\ln h\left( y_{i}\right) , \end{aligned}$$
(8)

respectively.

From (8), one can easily obtain the ML estimators of \(\tau _{1}\) and \(\tau _{2}\) by partially differentiating it with respect to \(\tau _{1}\) and \(\tau _{2}\); and equating these differential equations to zero. After obtaining the ML estimators of \(\tau _{1}\) and \(\tau _{2}\), the ML estimator of \({R}_{s,k}\) can be obtained by utilizing the invariance property of the ML estimators. Hence, the ML estimators of \(\tau _{1}\), \(\tau _{2}\) and \(R_{s,k}\) are obtained as follows:

\(\tilde{\tau _{1}}=-nk\left[\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{k} \ln \left[1-H\left( x_{ij}\right) \right] \right] ^{-1}\), \(\tilde{\tau _{2}}=-n\left[\displaystyle \sum _{i=1}^{n} \ln \left[H\left( y_{i}\right) \right] \right] ^{-1}\)

and

$$\begin{aligned} {\tilde{R}}_{s,k}=\Psi _1\tilde{\tau }_{2} B\left( \tilde{\tau }_{2},\tilde{\tau _{1}}(i+j)+1\right) , \end{aligned}$$
(9)

respectively.

3.1 Fisher Information Matrix and Asymptotic Confidence Interval

The Fisher information matrix of \(\underline{\tau }=(\tau _{1},\tau _{2})\) is given by

$$\begin{aligned} I(\underline{\tau })=\left( \begin{array}{cc} nk\tau _{1}^{-2} &{} 0 \\ 0 &{} n\tau _{2}^{-2}\end{array}\right) . \end{aligned}$$

Now, it is obvious that \(R_{s,k} \sim N\left( R_{s,k}, V({\tilde{R}}_{s,k})\right)\) asymptotically, where \(V({\tilde{R}}_{s,k})=\sum _{i=1}^{2}\sum _{j=1}^{2}\dfrac{\partial R_{s,k}}{\partial \tau _{i}}\dfrac{\partial R_{s,k}}{\partial \tau _{j}}I^{-1}_{ij}\) and \(I^{-1}_{ij}\) is the \((i,j)^{th}\) element of the inverse of I {see [27]}.

Here,

$$\begin{aligned} V({\tilde{R}}_{s,k})&=\left( \dfrac{\partial R_{s,k}}{\partial \tau _{1}}\right) ^2 I^{-1}_{11}+ 2 \dfrac{\partial R_{s,k}}{\partial \tau _{1}}\dfrac{\partial R_{s,k}}{\partial \tau _{2}}I^{-1}_{12}+\left( \dfrac{\partial R_{s,k}}{\partial \tau _{2}}\right) ^2 I^{-1}_{22},\\&=\left( \dfrac{\partial R_{s,k}}{\partial \tau _{1}}\right) ^2 \tau _{1}^2\left( nk\right) ^{-1}+ \left( \dfrac{\partial R_{s,k}}{\partial \tau _{2}}\right) ^2 \tau _{2}^2 n^{-1}, \end{aligned}$$

\(\dfrac{\partial R_{s,k}}{\partial \tau _{1}}=\Psi _1\left( i+j\right) \tau _{2}B( \tau _{2},\tau _{1}(i+j)+1)\left[\psi (\tau _{1}(i+j)+1)-\psi (\tau _{2}+\tau _{1}(i+j)+1)\right]\),

\(\dfrac{\partial R_{s,k}}{\partial \tau _{2}}=\Psi _1\tau _{2}B(\tau _{2},\tau _{1}(i+j)+1)\left[\psi (\tau _{2}+1)-\psi (\tau _{2}+\tau _{1}(i+j)+1)\right]\)

and

\(\psi (x)=\dfrac{\ln \left( \Gamma (x)\right) }{dx}=\dfrac{\Gamma '(x)}{\Gamma (x)}\) is the digamma function.

Therefore, \(100(1-\alpha)\%\) ACI of \(R_{s,k}\) is given by

$$\begin{aligned} R_{s,k}\in {\tilde{R}}_{s,k} \pm z_{\alpha /2}\sqrt{{\tilde{V}}\left( {\tilde{R}}_{s,k}\right) }, \end{aligned}$$
(10)

where \(z_{\alpha /2}\) is the upper \(\alpha /2{th}\) quantile of the standard normal distribution and \(\sqrt{{\tilde{V}}\left( {\tilde{R}}_{s,k}\right) }=\left. \displaystyle \sqrt{V\left( {{\tilde{R}}}_{s,k}\right) }\right| _{\{\tilde{\tau }_{1},\tilde{\tau }_{2}\}}\).

4 Bayesian Estimation of \(R_{s,k}\)

This section deals with the Bayesian estimation of \(R_{s,k}\). Selection of prior distributions plays a very important role for derivation of Bayes estimator. Ref. [28] evince that in Bayesian estimation problems there is no clear method on choosing priors are available. To obtain the Bayes estimator of \(R_{s,k}\), two independent gamma priors for the parameters \(\tau _{1}\) and \(\tau _{2}\) are considered. The main motivation behind the selection of gamma prior are; (i) the positive support for the parameters \(\tau _{1}>0\) and \(\tau _{2}>0\) of the considered models and (ii) gamma class of priors are conjugate priors. Refs. [6, 23] used the gamma priors in the estimation of the reliability function \(R_{s,k}\) and they have shown that the gamma priors are highly flexible in nature and also become the obvious choice in case of conjugate priors. Assuming that, \(\tau _{1} \sim Gamma(p_1,q_1)\) and \(\tau _{2} \sim Gamma(p_2,q_2)\), the joint posterior density of \((\tau _{1},\tau _{2})\) comes out to be

$$\begin{aligned}&\displaystyle \pi (\tau _{1},\tau _{2}|X,\underline{y}) \propto \tau _{1}^{nk+p_1-1}\tau _{2}^{n+p_2-1}e^{(\tau _{1}-1)\sum _{i=1}^{n}\sum _{j=1}^{k}\ln \left[1-H\left( x_{ij}\right) \right] }\\&\quad .e^{(\tau _{2}-1)\sum _{i=1}^{n}\ln \left[H( y_{i})\right] } e^{-q_1\tau _{1}}e^{-q_2\tau _{2}}. \end{aligned}$$

Hence, the marginal posteriors of \(\tau _{1}\) and \(\tau _{2}\) are

$$\begin{aligned} \displaystyle \pi (\tau _{1}) \propto Gamma\left( nk+p_1, ~q_1-\sum _{i=1}^{n}\sum _{j=1}^{k}\ln \left[1-H\left( x_{ij}\right) \right] \right) \end{aligned}$$
(11)

and

$$\begin{aligned} \displaystyle \pi (\tau _{2}) \propto Gamma\left( n+p_2, ~q_2-\sum _{i=1}^{n}\ln \left[H\left( y_{i}\right) \right] \right) . \end{aligned}$$
(12)

Now, \(R_{s,k}^{MCMC}\) is computed as follows:

  1. 1.

    Set l=1.

  2. 2.

    Generate \(\tau _{1}^{(l)}\) from (11).

  3. 3.

    Generate \(\tau _{2}^{(l)}\) from (12).

  4. 4.

    Compute \(R_{s,k}^{(l)}=\Psi _1\tau _{2}^{(l)} B(\tau _{2}^{(l)},\tau _{1}^{(l)}(i+j)+1)\).

  5. 5.

    Set l=l+1.

  6. 6.

    Repeat steps 2 to 5 N number of times and get the posterior sample \(R_{s,k}^{(l)}\), \(l=1,2,...,N.\)

Then, under squared error loss function (SELF) \(R_{s,k}^{MCMC}\) is comes out to be {see [29]}

$$\begin{aligned} {R}_{s,k}^{MCMC}=\dfrac{1}{N-M_1}\sum _{l=M_1+1}^{N-M_1} R_{s,k}^{(l)}. \end{aligned}$$
(13)

5 ML Estimation of \(R_{s,k}\), when Strength Population is PRHR Model and Stress Population is PHR Model

This section deals with the ML estimation of \(R_{s,k}\), when the stress variate follow the PHR model and strength variates follows the PRHR model. Also, it is assumed that \(Y, X_i,~i=1,2,...,k\) are independent. Let PRHR(\(\tau _{2}\)) denotes the common pdf of the independent strength variates, \(X_i,~i=1,2,...,k\) and PHR(\(\tau _{1}\)) denotes the pdf of Y. Then by using (1), (2) and (4) the reliability in multicomponent stress–strength set-up is obtained as

$$\begin{aligned} R_{s,k}&=\displaystyle \sum _{i=s}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) \tau _{1}\int _{0}^{\infty } \left[1-\left[H( y)\right] ^{\tau _{2}}\right] ^{i}\left[H( y)^{\tau _{2}}\right] ^{k-i} \left[1-H( y)\right] ^{\tau _{1}-1} h( y)dy\nonumber \\&=\Psi _2\tau _{1}\int _{0}^{\infty } [H( y)]^{\tau _{2} (k-i+j)}\left[1-H( y)\right] ^{\tau _{1}-1} h( y)dy,\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text {where}~ \Psi _2=\displaystyle \sum _{i=s}^{k}\sum _{j=0}^{i}\left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}i\\ j\end{array}}\right) (-1)^{j}\nonumber \\&=\Psi _2\tau _{1}\int _{0}^{1} z^{\tau _{2}\left( k-i+j\right) }\left( 1-z\right) ^{\tau _{1}-1}dz,~\text {where}~z=H( y)\nonumber \\&=\Psi _2\tau _{1} B\left( \tau _{2}(k-i+j)+1,\tau _{1}\right) . \end{aligned}$$
(14)

Let us assume that n systems are put on life testing experiment from strength and stress populations, respectively and under this set-up the following data is observed: \(X_{ij}\) and \(Y_{i}\), \(i=1,2,...,n\), \(j=1,2,...,k\). Thus, the likelihood and log-likelihood functions corresponding to the above data set are given by

$$\begin{aligned} L_2(\tau _{1},\tau _{2}|X,\underline{y})=\prod _{i=1}^{n}\left( \prod _{j=1}^{k} f_{X}(x_{ij};\tau _{2})\right) f_{Y}(y_{i};\tau _{1}), \end{aligned}$$
(15)

and

$$\begin{aligned} l_2(\tau _{1},\tau _{2}|X,\underline{y})&=nk\ln \tau _{2}+n\ln \tau _{1}+(\tau _{2}-1) \displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{k} \ln H( x_{ij})\nonumber \\&\quad +\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{k}\ln h( x_{ij})+\left( \tau _{1}-1\right) \displaystyle \sum _{i=1}^{n} \ln \left[1-H( y_{i})\right] +\displaystyle \sum _{i=1}^{n}\ln h( y_{i}), \end{aligned}$$
(16)

respectively.

Now, proceeding on the similar lines as in Sect. 3, the ML estimators of \(\tau _{1}\), \(\tau _{2}\) and \(R_{s,k}\) are obtained using (16) as follows:

\(\tilde{\tau _{1}}=-n\left[\displaystyle \sum _{i=1}^{n} \ln \left[1-H( y_{i})\right] \right] ^{-1}\), \(\tilde{\tau _{2}}=-nk\left[\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{k} \ln \left[H( x_{ij})\right] \right] ^{-1}\)

and

$$\begin{aligned} {\tilde{R}}_{s,k}=\Psi _2\tilde{\tau _{1}} B\left( \tilde{\tau _{2}}(k-i+j)+1,\tilde{\tau _{1}}\right) , \end{aligned}$$
(17)

respectively.

5.1 Fisher Information Matrix and Asymptotic Confidence Interval

The Fisher information matrix of \(\underline{\tau }=(\tau _{1},\tau _{2})\) is given by

$$\begin{aligned} I(\underline{\tau })=\left( \begin{array}{cc} n\tau _{1}^{-2} &{} 0 \\ 0 &{} nk\tau _{2}^{-2}\end{array}\right) . \end{aligned}$$

Now, it is obvious that \(R_{s,k} \sim N\left( R_{s,k}, V({\tilde{R}}_{s,k})\right)\) asymptotically, where \(V({\tilde{R}}_{s,k})=\sum _{i=1}^{2}\sum _{j=1}^{2}\dfrac{\partial R_{s,k}}{\partial \tau _{i}}\dfrac{\partial R_{s,k}}{\partial \tau _{j}}I^{-1}_{ij}\) and \(I^{-1}_{ij}\) is the \((i,j)^{th}\) element of the inverse of I.

Here,

$$\begin{aligned} V({\tilde{R}}_{s,k})=\left( \dfrac{\partial R_{s,k}}{\partial \tau _{1}}\right) ^2 \tau _{1}^2 n^{-1} +\left( \dfrac{\partial R_{s,k}}{\partial \tau _{2}}\right) ^2 \tau _{2}^2\left( nk\right) ^{-1}. \end{aligned}$$

\(\dfrac{\partial R_{s,k}}{\partial \tau _{1}}=\Psi _2\tau _{1}B(\tau _{1}, \tau _{2}(k-i+j)+1)\left[\psi (\tau _{1}+1)-\psi (\tau _{1}+\tau _{2}(k-i+j)+1)\right]\),

\(\dfrac{\partial R_{s,k}}{\partial \tau _{2}}=\Psi _2\left( k-i+j\right) \tau _{1}B(\tau _{1},\tau _{2}\left( k-i+j\right) +1)\left[\psi (\tau _{2}(k-i+j)+1)-\psi (\tau _{2}(k-i+j)+\tau _{1}+1)\right]\).

Therefore, \(100(1-\alpha)\%\) ACI of \(R_{s,k}\) is given by

$$\begin{aligned} R_{s,k}\in {\tilde{R}}_{s,k} \pm z_{\alpha /2}\sqrt{{\tilde{V}}\left( {\tilde{R}}_{s,k}\right) }. \end{aligned}$$
(18)

where \(\sqrt{{\tilde{V}}\left( {\tilde{R}}_{s,k}\right) }=\left. \displaystyle \sqrt{V\left( {{\tilde{R}}}_{s,k}\right) }\right| _{\{\tilde{\tau }_{1},\tilde{\tau }_{2}\}}\).

6 Bayesian Estimation of \(R_{s,k}\)

For obtaining the Bayes estimator of \(R_{s,k}\), we have again considered the two independent gamma priors given in Sect. 4 for \(\tau _{1}\) and \(\tau _{2}\). Using these gamma priors the joint posterior density of \((\tau _{1},\tau _{2})\) comes out to be

$$\begin{aligned}&\displaystyle \pi (\tau _{1},\tau _{2}|X,\underline{y}) \propto \tau _{1}^{n+p_1-1}\tau _{2}^{nk+p_2-1}e^{(\tau _{1}-1)\sum _{i=1}^{n}\ln \left[1-H( y_{i})\right] }\nonumber \\&\quad .e^{(\tau _{2}-1)\sum _{i=1}^{n}\sum _{j=1}^{k}\ln \left[H( x_{ij})\right] } e^{-q_1\tau _{1}}e^{-q_2\tau _{2}}. \end{aligned}$$

Hence, the marginal posterior distributions of \(\tau _{1}\) and \(\tau _{2}\) are

$$\begin{aligned} \displaystyle \pi (\tau _{1}) \propto Gamma\left( n+p_1, ~q_1-\sum _{i=1}^{n}\ln \left[1-H( y_{i})\right] \right) \end{aligned}$$
(19)

and

$$\begin{aligned} \displaystyle \pi (\tau _{2}) \propto Gamma\left( nk+p_2, ~q_2-\sum _{i=1}^{n}\sum _{j=1}^{k}\ln \left[H( x_{ij})\right] \right) . \end{aligned}$$
(20)

Now, \(R_{s,k}^{MCMC}\) is computed as follows:

  1. 1.

    Set l=1.

  2. 2.

    Generate \(\tau _{1}^{(l)}\) from (19).

  3. 3.

    Generate \(\tau _{2}^{(l)}\) from (20).

  4. 4.

    Compute \(R_{s,k}^{(l)}=\Psi _2\tau _{1}^{(l)} B(\tau _{2}^{(l)}(k-i+j)+1,\tau _{1}^{(l)}).\)

  5. 5.

    Set l=l+1.

  6. 6.

    Repeat steps 2 to 5, N number of times and get the posterior sample \(R_{s,k}^{(l)}\), \(l=1,2,...,N.\)

Then, the Bayes estimate of \(R_{s,k}\) under SELF comes out to be

$$\begin{aligned} {R}_{s,k}^{MCMC}=\dfrac{1}{N-M_2}\sum _{l=M_2+1}^{N-M_2} R_{s,k}^{(l)}. \end{aligned}$$
(21)

7 Simulation Study

In this section, particular cases of each of PHR and PRHR models are considered. It can be seen that, for \(H(x)=1-exp(-x)\), the exponential distribution comes out to be a particular case of PHR model and exponentiated exponential distribution comes out to be a particular case of PRHR model with following cdfs:

$$\begin{aligned} F_X(x)=1-exp\left( -\tau _{1}x\right) ; x>0,\tau _{1}>0, \end{aligned}$$
(22)

and

$$\begin{aligned} G_Y(y)=\left( 1-exp\left( -y\right) \right) ^{\tau _{2}}; y>0,\tau _{2}>0, \end{aligned}$$
(23)

respectively, where \(\tau _{1}\) and \(\tau _{2}\) are the parameters of the distributions.

To carry out the simulation study, both the cases discussed earlier in this paper, are considered one by one. Monte Carlo simulation and the MCMC methods are used, and comparison of the obtained estimators is done on the basis of mean square error (MSE). To compute the MCMC estimates, MCMC chain of 10,000 iterations are performed. First 3000 observations are discarded to diminish the effect of the starting distribution and remaining observations are considered for the computational purpose.

Under case-I, exponential distribution is considered as a particular case of PHR(\(\tau _{1}\)) model for the common pdf of the independent strength variates and exponentiated exponential distribution is considered as a particular case of PRHR(\(\tau _{2}\)) model for the pdf of the stress variate. Now, 1000 random samples each of sizes n=10, 20, 30, 50, 70 and 100, corresponding to each strength variate from PHR(\(\tau _{1}\)) and stress variate from PRHR(\(\tau _{2}\)) models are generated. These samples are generated for different values of the parameters and (s, k). For \((\tau _{1},\tau _{2})\)=(2,1), (2,3), (1,2), (2,2) and (s, k)=(1,4), (2,5), the average of ML and MCMC estimates (using priors 1, 2 and 3) of \(R_{s,k}\) are computed. The obtained average of estimates with their corresponding MSEs in parentheses are reported in Table 1. Furthermore, for these samples the ACIs with corresponding length of the intervals are also computed and presented in Table 3.

The MCMC estimates are obtained using the following three priors:

  • Prior 1: \(p_1=1;~ p_2=1;~ q_1=1;~ q_2=1;\)

  • Prior 2: \(p_1=1;~ p_2=1;~ q_1=0;~ q_2=0;\)

  • Prior 3: \(p_1=2;~ p_2=2;~ q_1=2;~ q_2=2.\)

Now, under case-II, exponentiated exponential distribution is considered as a particular case of PRHR(\(\tau _{2}\)) model for the common pdf of the independent strength variates and exponential distribution is considered as a particular case of PHR(\(\tau _{1}\)) model for the pdf of the stress variate. Proceeding on the similar lines as in Case-I, 1000 random samples each of sizes n=10, 20, 30, 50, 70 and 100 are generated corresponding to each strength and stress variates. These samples are generated for different values of the parameters and (s, k). To avoid the biasness in the selection of the parameters, different set of parameters, \((\tau _{1},\tau _{2})\)=(1,2), (3,1), (1,1) and (1,0.5) are considered. For these parameter values and (s, k)=(2,4) and (1,5), the average of ML and MCMC estimates (using priors 1, 2 and 3) of \(R_{s,k}\) are computed along with their corresponding MSEs in parentheses and these results are reported in Table 2. ACIs with their corresponding length of the intervals are also computed and presented in Table 4.

Here, it should be noted that the selection of the parameters \((\tau _{1},\tau _{2})\) is done in such a way that the values of \(R_{s,k}\) will vary from lower to moderate and higher. The performances of the estimators are also compared for different values of \(R_{s,k}\). For example, \(R_{2,5}\)=0.1245 {see Table 1}, shows that the probability that at least 2 out of 5 strength components exceed stress is very low, i.e., probability of the survival of the system is very low when two or more components survives the stress. On the other hand, \(R_{1,4}\)=0.6667 {see Table 1}, shows that there are good chances of survival of the system if one or more components survives the stress. Moreover, from Tables 1 and 2, it is also clear that the survival probability of the system, very much depends on the values of \((\tau _{1}, \tau _{2})\).

From Tables 1 and 2, it can also be observed that the computed MSEs using MCMC method are lower than that of ML estimators. But an improvement in the performances of the ML estimators can be observed as the sample size increases. For the large values of n all the estimators are efficiently performing. Also, it can be seen that for lower values of the sample sizes and higher values of the multicomponent reliabilities viz., 0.8333, 0.9821 and so on the MCMC estimators under prior 2 perform better than other MCMC estimators. Furthermore, for moderate values of multicomponent reliabilities the MCMC estimator under prior 3 performs quite well. Better performance of MCMC estimator under prior 2 is also observed for lower values of \(R_{s,k}\) as well.

In addition to this, one may observe from Tables 3 and 4 that the length of the ACI decreases as the sample size increases. The behaviour of the length of the ACIs can also be observed for different values of the parameters and (s, k).

Table 1 Estimates of \(R_{s,k}\), when strength variates follow PHR model (Exponential distribution) and stress variate follows PRHR model (Exponentiated exponential distribution)(CASE I)
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Table 2 Estimates of \(R_{s,k}\), when strength variates follow PRHR model (Exponentiated exponential distribution) and stress variate follows PHR model (Exponential distribution)(CASE II)
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Table 3 ACI of \(R_{s,k}\), when strength variates follow PHR model (Exponential distribution) and stress variate follows PRHR model (Exponentiated exponential distribution)(CASE I)
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Table 4 ACI of \(R_{s,k}\), when strength variates follow PRHR model (Exponentiated exponential distribution) and stress variate follows PHR model (Exponential distribution)(CASE II)
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8 Real Data Study

In this section, we consider the data discussed by [30]. They used this data to fit two parameter unit-Gompertz distribution and compared the fitting with other distributions such as Log-Lindley, Kumaraswamy and unit-Gamma distribution. Ref. [30] fitted the same distribution on strength as well as stress data. But, in our study we have fitted different distributions, namely; exponential distribution and exponentiated exponential distribution on stress and strength populations and vice versa.

Ref. [30] reported in their study that this data set represents the time between failures (in seconds) and is also available in the book by [31]. This failure time data is arranged into k=4-components and n=7-systems, by taking \(Y_1\) as the second failure time and \(X_{1k},~k=1,2,3,4\) be the failure times of observations numbered from 3 to 6. Similarly, remaining elements were arranged and a multicomponent system is obtained as follows:

\(X:\left( \begin{array}{cccc} 4&{} 36&{} 4&{} 5\\ 91&{} 49&{} 1&{} 25\\ 4&{} 30&{} 42&{} 9\\ 44&{} 32&{} 3&{} 78\\ 30&{} 205&{} 5&{} 129\\ 224&{} 186&{} 53&{} 14\\ 2&{} 10&{} 1&{} 34 \end{array} \right) ~~~Y:\left( \begin{array}{c} 10\\ 4\\ 1\\ 49\\ 1\\ 103\\ 9 \end{array} \right)\)

By using the transformation \(X=X/largest(X)\) and \(Y=Y/largest(Y)\), we get the following data:

\(X:\left( \begin{array}{cccc} 0.018&{} 0.161&{} 0.018&{} 0.022\\ 0.406&{} 0.219&{} 0.004&{} 0.112\\ 0.018&{} 0.134&{} 0.188&{} 0.040\\ 0.196&{} 0.143&{} 0.013&{} 0.348\\ 0.134&{} 0.915&{} 0.022&{} 0.576\\ 1.000&{} 0.830&{} 0.237&{} 0.062\\ 0.009&{} 0.045&{} 0.004&{} 0.152 \end{array} \right) ~~~Y:\left( \begin{array}{c} 0.097\\ 0.039\\ 0.010\\ 0.476\\ 0.010\\ 1.000\\ 0.087 \end{array} \right)\)

Now, we fit the exponential distribution and exponentiated exponential distribution on the transformed data. From, this data, it is clear that n=7 and k=4. For different values of s, we have computed the estimates of multicomponent reliability.

Here, both the cases are considered, one by one. Firstly, let us consider the first case in which the strength data follows the exponential distribution with parameter \(\tau _{1}\) and stress data follows the exponentiated exponential distribution with parameter \(\tau _{2}\). Thereafter, the ML estimates of multicomponent reliability are obtained at \(\left( \tilde{\tau _{1}},\tilde{\tau _{2}}\right)\) and the MCMC estimates are obtained under Priors 1, 2 and 3. The results for fitting the strength data X to the exponential distribution and Y data to the exponentiated exponential distribution are presented in Table 5 and corresponding multicomponent reliability estimates are reported in Table 6.

Table 5 Real data fitting in Case I
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Table 6 ML and MCMC estimates of \(R_{s,k}\) under Case I
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Further, for case I, the trace plot of the MCMC estimates under priors 1, 2 and 3 with \((n,k,s)=(7,4,1)\), are shown in Fig. 1.

Fig. 1
figure 1

Trace plots for Case I

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Let us consider the second case in which the strength data follows the exponentiated exponential distribution with parameter \(\tau _{2}\) and stress data follows the exponential distribution with parameter \(\tau _{1}\). The results for fitting the strength data X to the exponentiated exponential distribution and Y data to the exponential distribution are presented in Table 7 and corresponding multicomponent reliability estimates are reported in Table 8.

Table 7 Real data fitting in Case II
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Table 8 ML and MCMC estimates of \(R_{s,k}\) under Case II
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For case II, the trace plot of the MCMC estimates under priors 1-3 with \((n,k,s)=(7,4,1)\) are shown in Fig. 2.

Fig. 2
figure 2

Trace plots for Case II

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9 Discussions and Conclusions

Before conceptualizing the paper, we have gone through the various literatures and papers on multicomponent reliability, but we did not find enough literature on multicomponent reliability in which stress and strength variables follow different distributions. But in the real life situations, it is very much possible that the stress and strength variables may follow different lifetime distributions. To handle such situations, we have developed the estimation procedures for the multicomponent reliability under the assumption that the strength variates follow PHR model and stress variate follows PRHR model and vice versa.

In this article, we have considered the both classical and Bayesian inferences for the estimation of the reliability function in multicomponent set-up. We found that the Bayes estimates cannot be obtained in explicit form. We used the MCMC technique to compute the approximate Bayes estimates. Furthermore, we compared the Bayes estimates based on gamma priors with the corresponding ML estimates and found that the behavior of MCMC estimates very much depends on the values of the prior parameters, as one may expect. Moreover, On the basis of Tables 1, 2, 3 and 4 the following observations are made:

  1. 1.

    As the sample size n increases, the MSEs corresponding to the various estimates decrease.

  2. 2.

    For large values of n, the performance of ML and MCMC estimators are quite similar.

  3. 3.

    For small to moderate sample sizes, the performance of MCMC estimates are better than the ML estimates.

  4. 4.

    The length of the ACI decreases as sample size increases.

In the present article, the justification of the estimation procedures is done through the simulation studies and the utility of the article is shown through real data example. Furthermore, we have seen in the simulation study that for moderate values of multicomponent reliabilities, MCMC estimators under prior 3 gives better result. In the view of this discussion, one can conclude that the MCMC estimate under prior 3 is more accurate in case of real data example.