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An Optimal Bayesian Sampling Plan for Two-Parameter Exponential Distribution Under Type-I Hybrid Censoring

Abstract

The Bayesian sampling plan for two-parameter exponential distribution has been considered by Lam (Statistician, 39, 53–66, 1990) under the conventional Type-II censoring. Lin et al. (Commun. Stat.—Simul. Comput., 37, 1101–1116, 2008b) have obtained an exact Bayesian sampling plan for one-parameter exponential distribution under Type-I and Type-II hybrid censoring schemes. In this paper, we obtain an optimal Bayesian sampling plan for the two-parameter exponential distribution under Type-I hybrid censoring scheme based on a four-parameter conjugate prior, introduced by Varde (J. Am. Stat. Assoc., 64, 621–631 1969). Bayes risk expressions of Lam (Statistician, 39, 53–66, 1990) for the conventional Type-II censoring scheme can be obtained as special cases of the Type-I hybrid censoring scheme. The optimal Bayesian sampling plan cannot be obtained analytically, we provide a numerical algorithm to compute the optimal Bayesian sampling plan. Different optimal Bayesian sampling plans have been reported.

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Acknowledgments

The authors would like to thank the unknown reviewers for their helpful comments which have significantly improved the quality of the manuscript. We also would like to thank to High Performance Computing (HPC) systems at Computer Center, Indian Institute of Technology Kanpur.

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Correspondence to Sharmishtha Mitra.

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Appendices

Appendix A: Proof of Theorem 3.2

$$ \begin{array}{@{}rcl@{}} r(n,r,T,\xi) &= & E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}\left[ \ell(\delta(\underline{X}),\lambda,\mu,n,r,T) \right] \\ &= &\! E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu} \left[ nC_{s}+\tau C_{T}-(n-D) r_{s}+\delta (\underline{X})\left(\sum\limits_{0 \leq i+j \leq 2}{C_{ij} \lambda^{i} \mu^{j}}\right) + (1- \delta (\underline{X}))C_{r}\right] \\ &= & n(C_{s}-r_{s})+C_{T} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu)+ r_{s} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) + C_{r} \\ && + E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}\left[\delta(\underline{X})\left( \sum\limits_{0 \leq i+j \leq 2}{C_{ij} \lambda^{i} \mu^{j}}-C_{r}\right)\right] \end{array} $$
(A.1)

Equation A.1 further can be simplified as follows

$$ \begin{array}{@{}rcl@{}} r(n,r,T,\xi) &= & n(C_{s}-r_{s})+C_{r}+C_{T} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu)+ r_{s} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) \\ && + E_{\lambda,\mu} \left[\left( \sum\limits_{0 \leq i+j \leq 2}{C_{ij} \lambda^{i} \mu^{j}}-C_{r} \right) \left( P(D=0)+P(\widehat{W}\geq \xi |\lambda,\mu)\right)\right] \\ &= & n(C_{s}-r_{s})+C_{r}+C_{T} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu)+ r_{s} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) \\ && + (C_{00}-C_{r}) E_{\lambda,\mu} \left(e^{-n \lambda (T-\mu)}\right) + \sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}\left( \lambda^{i} \mu^{j} e^{-n \lambda (T-\mu )} \right) \\&&+ \sum\limits_{0 \leq i+j \leq 2}{C^{\prime}_{ij} I_{ij} }, \end{array} $$

where \( I_{ij} = E_{\lambda ,\mu } \left ( \lambda ^{i} \mu ^{j} P(\widehat {W}\geq \xi |\lambda ,\mu ) \right ) \). Further \( E_{\lambda ,\mu } E_{\underline {X}|\lambda ,\mu }(\tau |\lambda ,\mu ) \), Eλ,μ \(E_{\underline {X}|\lambda ,\mu }(D|\lambda ,\mu )\) and \( {\sum }_{0 \leq i+j \leq 2} C_{ij} E_{\lambda ,\mu }\left (\lambda ^{i} \mu ^{j} e^{-n \lambda (T-\mu )} \right )\) and Iij have been obtained. In the remaining proof, it is to be noted that the Bayes risk in explicit forms has been provided for α > 1.

Computation of I ij

Case (i)::

r = 1 and n ≥ 1

Suppose notations η1, \(\xi _{\mu }^{*}\) and \(\xi _{T}^{*}\) are defined as \(\eta _{1} = {\min \limits } \{\eta ,T\}\), \(\xi _{\mu }^{*}= \max \limits \{\xi ,\mu \}\) and \(\xi _{T}^{*}= \max \limits \{\xi ,T\}\), respectively, then

$$ \begin{array}{@{}rcl@{}} I_{ij} & =& \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} f_{\widehat{W}}(x) \ dx d\lambda d\mu \\ & =& \frac{1}{A} \sum\limits_{l=0}^{\infty} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{nl} \left\{ g(x-\mu;1,n\lambda)\right.\\ &&\left.- q^n g(x-T;1,n\lambda) \right\} \ dx d\lambda d\mu \\ & =& \frac{1}{A} \sum\limits_{l=0}^{\infty} \left\{ {\int}_{0}^{\eta_1} {\int}_{\xi_{\mu}^{*}}^{\infty} {\int}_{0}^{\infty} n \lambda^{\alpha+i-1} \mu^j e^{-\lambda \left( \beta -\gamma \mu + nl(T-\mu) \right)} \left( n \lambda e^{-n \lambda (x-\mu)} \right) \ d\lambda dx d\mu \right.\\ &&\left.- {\int}_{0}^{\eta_1} {\int}_{\xi_{T}^{*}}^{\infty} {\int}_{0}^{\infty} n \lambda^{\alpha+i-1} \mu^j e^{-\lambda \left( \beta -\gamma \mu + n(l+1)(T-\mu) \right)} \left( n \lambda e^{-n \lambda (x-T)} \right) \ d\lambda dx d\mu \right\} \\ & =& \frac{1}{A} \sum\limits_{l=0}^{\infty} \left\{ {\int}_{0}^{\eta_1} {\int}_{\xi_{\mu}^{*}}^{\infty} \frac{n {\Gamma}(\alpha+i+1) \mu^j }{\left( \beta -\gamma \mu + nl(T-\mu) + n(x-\mu) \right)^{\alpha+i+1}} dx d\mu \right. \\ && \left.- {\int}_{0}^{\eta_1} {\int}_{\xi_{T}^{*}}^{\infty} \frac{n {\Gamma}(\alpha+i+1) \mu^j }{\left( \beta -\gamma \mu + n(l+1)(T-\mu) + n(x-T) \right)^{\alpha+i+1}} dx d\mu \right\} \\ & =& \frac{\Gamma{(\alpha+i)}}{A} \sum\limits_{l=0}^{\infty} \left\{ {\int}_{0}^{\eta_1} \frac{\mu^j }{\left( \beta -\gamma \mu + nl(T-\mu) + n(\xi_{\mu}^{*} -\mu) \right)^{\alpha+i}} \ d\mu \right. \\ &&\left. - {\int}_{0}^{\eta_1} \frac{\mu^j }{\left( \beta -\gamma \mu + n(l+1)(T-\mu) + n(\xi_{T}^{*} -T) \right)^{\alpha+i}} \ d\mu \right\}. \end{array} $$
Case (ii)::

r = 2 and n = 2

In this case, Iij can be expressed as

$$ \begin{array}{@{}rcl@{}} I_{ij} &= & \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} P_{\underline{X}|\lambda,\mu}(\widehat{W}\geq \xi |\lambda,\mu) \ d\lambda d\mu \\ &= & \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} \left[ I_{\{ \xi < \mu \}} + I_{\{\mu \leq \xi \leq T \}} \left\{ \sum\limits_{l = 0}^{\infty} q^{2l} \left[ 2 q (1-q) S_1 (\xi) \right. \right.\right.\\ &&\left.\left.\left. + S_2 (\xi-\mu; 2, 2\lambda) + q^2 S_2 (\xi-T; 2, 2\lambda) - 2q S_2 (\xi-\frac{\mu+T}{2}; 2, 2\lambda) \right] \right\} \right] \ d\lambda d\mu \\ &= & \frac{1}{A} \left\{ M_0 + \sum\limits_{l=0}^{\infty} 2(M_{l1} - M_{l4}) + M_{l2} + M_{l3} \right\}, \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} M_{0} = {\int}_{\xi}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} \ d\lambda d\mu = {\Gamma}(\alpha+i) {\int}_{\xi}^{\eta_1} \frac{\mu^j}{(\beta-\gamma \mu )^{\alpha+i}} \ d \mu, \end{array} $$

and also, with another notation \(\eta _{2} = \min \limits \{\xi ,\eta _{1}\}\),

$$ \begin{array}{@{}rcl@{}} M_{l1} & = & {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{2l+1} (1-q) S_1(\xi) \ d\lambda d\mu \\ & = & \left\{\begin{array}{lll} {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda \left( \beta-\gamma \mu + (2l+1)(T-\mu)\right)} \left( 1-e^{-\lambda(T-\mu)}\right) \ d\lambda d\mu , & \text{if~} \xi < T \\ 0, & \text{if~} \xi \geq T \end{array}\right. \\ & = & \left\{\begin{array}{lll} {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda \left( \beta-\gamma \mu + (2l+1)(T-\mu)\right)} \ d\lambda d\mu \\ - {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda \left( \beta-\gamma \mu + 2(l+1)(T-\mu)\right)} \ d\lambda d\mu , & \text{if~} \xi < T \\ 0, & \text{if~} \xi \geq T \end{array}\right. \\ & = & \left\{\begin{array}{lll} {\Gamma}(\alpha + i) {\int}_{0}^{\eta_2} \mu^{j} \left\{ \frac{1}{\left( \beta-\gamma \mu + (2l+1)(T\!-\mu)\right)^{\alpha+i}} - \frac{1}{\left( \beta-\gamma \mu \!+ 2(l+\!1)(T\!-\mu)\right)^{\alpha+i}} \right\} \ d\mu , & \text{if~} \xi \!<\! T \\ 0, & \text{if~} \xi \!\geq\! T \end{array}\right. \end{array} $$

as α + i > 0, βγμ + (2l + 1)(Tμ) > 0 and βγμ + 2(l + 1)(Tμ) > 0,

$$ \begin{array}{@{}rcl@{}} M_{l2} &= & {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{2l} S_2(\xi-\mu,2,2\lambda) \ d\lambda d\mu \\ &= & {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu + 2l(T-\mu))} \left\{ \frac{(2\lambda)^2}{\Gamma(2)} {\int}_{\xi-\mu}^{\infty} u e^{-2 \lambda u} \ du \right\} d\lambda d\mu \\ &= & 4 {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} {\int}_{\xi-\mu}^{\infty} \lambda^{\alpha +i+1} \mu^{j} u e^{- \lambda (\beta-\gamma \mu + 2l(T-\mu)+2u)} \ du d\lambda d\mu. \end{array} $$

Note that βγμ + 2l(Tμ) + 2u is strictly positive, therefore

$$ \begin{array}{@{}rcl@{}} M_{l2} &= & 4 {\int}_{0}^{\eta_2} {\int}_{\xi-\mu}^{\infty} \frac{\Gamma(\alpha +i+2) \mu^{j} u }{ (\beta-\gamma \mu + 2l(T-\mu)+2u)^{\alpha+i+2}} \ du d\mu. \end{array} $$

Let x = 2u/(βγμ + 2l(Tμ)) and also define \(\xi _{1}^{*}(\mu ) = 2(\xi -\mu )/(\beta -\gamma \mu + 2l(T-\mu ))\), then

$$ \begin{array}{@{}rcl@{}} M_{l2} &= & {\Gamma}(\alpha +i) \left\{ \frac{1}{B(2,\alpha+i)} {\int}_{0}^{\eta_2} {\int}_{\xi_{1}^{*}(\mu)}^{\infty} \frac{\mu^{j}}{ (\beta-\gamma \mu + 2l(T-\mu))^{\alpha+i}} \frac{x^{2-1}}{(1+x)^{\alpha+i+2}} \ dx d\mu \right\} \\ &= & {\Gamma}(\alpha +i) D_{1}(2,\alpha+i,2l,\eta_2), \end{array} $$

where \(D_{s}(a,b,y,x) = {\int \limits _{0}^{x}} \frac {\mu ^{j} \left (1 - I_{\xi _{s}^{*}(\mu )/(1+\xi _{s}^{*}(\mu ))}(a,b)\right )}{ (\beta -\gamma \mu + y(T-\mu ))^{\alpha +i}} \ d\mu \) and It(a, b) denotes the cdf of probability distribution Beta(a,b) at t. Now, thus and so

$$ \begin{array}{@{}rcl@{}} M_{l3} &= & {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{2(l+1)} S_2(\xi-T,2,2\lambda) \ d\lambda d\mu \\ &= & \left\{\begin{array}{lll} {\Gamma}(\alpha +i) \int\limits_{0}^{\eta_2} \frac{\mu^{j}}{\left( \beta-\gamma \mu + 2(l+1)(T-\mu)\right)^{\alpha+i}} \ d\mu , & \text{if~} \xi < T \\ {\Gamma}(\alpha +i) D_{2}(2,\alpha+i,2(l+1),\eta_2), & \text{if~} \xi \geq T \end{array}\right. \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} M_{l4} &= & {\int}_{0}^{\eta_2} {\int}_{0}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{2l+1} S_2(\xi-\frac{\mu+T}{2},2,2\lambda) d \ d\lambda d\mu \\ &= & {\Gamma}(\alpha +i) \left\{ \int\limits_{\min\{\eta_2, \max\{0,2\xi-T\} \}}^{\eta_2} \frac{\mu^{j}}{ (\beta-\gamma \mu + (2l+1)(T-\mu))^{\alpha+i} } \ d\mu \right. \\ &&\left. +D_{3}(2,\alpha+i,2l+1,\max\{0,\min\{(2\xi-T),\eta_2\}\}) \vphantom{\int\limits_{min\{\eta_2, max\{0,2\xi-T\} \}}^{\eta_2}}\right\} \end{array} $$

with \(\xi _{2}^{*}(\mu ) = \frac {2(\xi -T)}{(\beta -\gamma \mu + 2(l+1)(T-\mu ))}\) and \( \xi _{3}^{*}(\mu ) = \frac {2(\xi -(\mu +T)/2)}{(\beta -\gamma \mu + (2l+1)(T-\mu ))}\), respectively.

Case (iii)::

r ≥ 2 and n ≥ 3

$$ \begin{array}{@{}rcl@{}} I_{ij} &= & \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} f_{\widehat{W}}(x) \ dx d\lambda d\mu \\ &= & \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} \left\{ \sum\limits_{l = 0}^{\infty} q^{nl} \left[ c_{10} g\left( -x+\mu+\mu_{10};1,\frac{\lambda}{n-2}\right) \right.\right.\\ && -e_{10} g\left( -x+T;1,\frac{\lambda}{n-2}\right) + I_{\{ r \geq 3\}} \sum\limits_{d=2}^{r-1}\sum\limits_{k=0}^{d-1} \left\{ c_{dk} g_{3}(x-\mu-\mu_{dk};d-1;\lambda d,\lambda_{dk}) \right. \\ &&\left. - e_{dk} g_{3}(x-T;d-1,\lambda d,\lambda_{dk}) \right\} + g_{1}(x-\mu;r-1,\lambda r,\lambda n)-q^{n}g_{1}(x-T;r-1,\lambda r,\lambda n) \\ &&\left.\left. + \sum\limits_{k=0}^{r-2} \left\{ c_{k} g_{3}(x-\mu-\mu_{k};r-1,\lambda r,\lambda_{k}) - e_{k}g_{3}(x-T;r-1,\lambda r,\lambda_{k}) \right\} \right] \right\} \ dx d\lambda d\mu. \end{array} $$

Define \( a_{dk} = \frac {d-k}{2d-k-n} \ \text {for} \ d = 2,3,, \dots ,r-1, \ k= 0,1,\dots ,d-1 \ \ni \ \lambda _{dk}= \lambda d a_{dk} \) and \( a_{k} = \frac {r-k-1}{2r-k-n-1} \ \text {for} \ k= 0,1,\dots ,r-1 \ \ni \ \lambda _{k}= \lambda r a_{k} \). Note that λdk > (<)0 ⇔ 2dkn > (<)0, \(\lambda _{dk} = \infty \iff 2d -k -n = 0\) and λk also behaves similarly. Therefore, define some sets as follows:

$$ \begin{array}{@{}rcl@{}} &&A_{0} = \{ (d,k) \ | \ d=2,3,\dots,r-1, k=0,1, \dots,d-1 \},\\ &&A_{1} = \{ (d,k) \ | \ (d,k) \in A_{0} \ \text{and} \ 2d-k-n > 0 \},\\ &&A_{2} = \{ (d,k) \ | \ (d,k) \in A_{0} \ \text{and} \ 2d-k-n < 0 \},\\ &&A_{3} = \{ (d,k) \ | \ (d,k) \in A_{0} \ \text{and} \ 2d-k-n = 0 \},\\ &&B_{0} = \{ k \ | \ k=0,1, \dots,r-2 \},\\ &&B_{1} = \{ k \ | \ k \in B_{0} \ \text{and} \ 2r-k-n-1 > 0 \},\\ &&B_{2} = \{ k \ | \ k \in B_{0} \ \text{and} \ 2r-k-n-1 < 0 \},\\ &&B_{3} = \{ k \ | \ k \in B_{0} \ \text{and} \ 2r-k-n-1 = 0 \}.\\ \end{array} $$

Now replace q, cdk, edk, ck, ek and g3 by their respective expressions to have

$$ \begin{array}{@{}rcl@{}} I_{ij} &= & \frac{1}{A} \sum\limits_{l=0}^{\infty} \left[ n(K_{l1}-K_{l2}) + I_{\{ r \geq 3\}} \left\{ \underset{A_1}{\sum\sum} (-1)^k \binom{n}{d} \binom{d}{k} (K_{l3}-K_{l4}) \right.\right.\\ &&\left. +\underset{A_2}{\sum\sum} (-1)^k \binom{n}{d} \binom{d}{k} (K_{l5}-K_{l6}) + \underset{A_3}{\sum\sum} (-1)^k \binom{n}{d} \binom{d}{k} (K_{l7}-K_{l8}) \right\} \\ && + (K_{l9}-K_{l10}) + \sum\limits_{B_1} (-1)^{k+1} r \binom{n}{r} \binom{r-1}{k} \frac{1}{n-r+k+1} (K_{l11}-K_{l12}) \\ && +\sum\limits_{B_2} (-1)^{k+1} r \binom{n}{r} \binom{r-1}{k} \frac{1}{n-r+k+1} (K_{l13}-K_{l14}) \\ && \left. + \sum\limits_{B_3} (-1)^{k+1} r \binom{n}{r} \binom{r-1}{k} \frac{1}{n-r+k+1} (K_{l15}-K_{l16}) \right], \end{array} $$

where \(K_{l1}, K_{l2},\dots , K_{l16}\) denotes some expressions involving triple integrals. Moreover these integrals can be obtained along the same line as Ml1 and Ml2 have been obtained, so their proofs have been omitted here.

$$ \begin{array}{@{}rcl@{}} K_{l1} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)-1} g(-x+\mu+\mu_{10};1,\lambda/(n-2)) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) {\int}_{0}^{\eta_1} \mu^{j} \left\{\frac{1}{\left( \beta-\gamma \mu + (n(l+1)-1)(T-\mu) \right)^{\alpha+i}} \right.\\ &&\left. - \frac{1}{\left( \beta-\gamma \mu + (n(l+1)-1)(T-\mu) - \frac{(\min\{\xi_{\mu}^{*}, \mu+\mu_{10}\}-\mu-\mu_{10})}{n-2} \right)^{\alpha+i}} \right\} \ d\mu, \\ K_{l2} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g(-x+T;1,\lambda/(n-2)) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) {\int}_{0}^{\eta_1} \mu^{j} \left\{\frac{1}{\left( \beta-\gamma \mu + n(l+1)(T-\mu) \right)^{\alpha+i}} \right.\\ && - \left.\frac{1}{\left( \beta-\gamma \mu + n(l+1)(T-\mu) - \frac{(\min\{\xi_{\mu}^{*}, T\}-T)}{n-2} \right)^{\alpha+i}} \right\} \ d\mu, \\ K_{l3} & = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)-d+k} g_1(x - \mu - \mu_{dk};d - 1,\lambda d, \lambda_{dk}) \ dx d\lambda d\mu \\ v&= & {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{d-2} R_{dks} D_{4}(d-s-1, \alpha+i, n(l+1)-d+k, \eta_1) + \frac{ 1 }{ (1-a_{dk})^{d-1} }\right. \\ && \left.\!\times\! {\int}_{0}^{\eta_1} \frac{ \mu^{j}}{ \left( \beta - \gamma \mu + (n(l + 1) - d + k)(T - \mu) + d a_{dk} (\max\{\xi, \mu+\mu_{dk}\}-\mu-\mu_{dk}) \right)^{\alpha+i } } d\mu \right\} \end{array} $$

where \(R_{dks} = (-1)^{s} \frac { a_{dk} }{ (a_{dk}-1)^{s+1} }\) and \(\xi _{4}^{*}(\mu ) = \frac {d(\max \limits \{\xi , \mu +\mu _{dk}\} - \mu - \mu _{dk})}{ \beta -\gamma \mu + (n(l+1)-d+k)(T-\mu ) }\),

$$ \begin{array}{@{}rcl@{}} K_{l4} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g_1(x-T;d-1,\lambda d, \lambda_{dk}) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{d-2} R_{dks} D_{5}(d-s-1, \alpha+i, n(l+1), \eta_1) \right. \\ &&\left. v + \frac{ 1 }{ (1-a_{dk})^{d-1} } {\int}_{0}^{\eta_1} \frac{ \mu^j }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) + d a_{dk} (\xi_{T}^{*}-T) \right)^{\alpha+i } } d\mu \right\} \end{array} $$

with \(\xi _{5}^{*}(\mu ) = \frac { d (\xi _{T}^{*}-T) }{(\beta -\gamma \mu + n(l+1)(T-\mu ))}\),

$$ \begin{array}{@{}rcl@{}} K_{l5} & = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{-\! \lambda (\beta-\gamma \mu)} q^{n(l + 1)-\!d+k} g_2(x - \mu - \mu_{dk};d - 1,\lambda d, - \lambda_{dk}) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha +i) \left[ \sum\limits_{s=0}^{d-2} R_{dks} D_4(d-s-1,\alpha+i,n(l+1)-d+k,\eta_1) + \frac{1}{(1-a_{dk})^{d-1}} \right. \\ && \times {\int}_{0}^{\eta_1} \mu^j \left\{ \frac{ 1 }{ \left( \beta-\gamma \mu + (n(l+1)-d+k)(T-\mu) \right)^{\alpha+i } } \right. \\ &&\left.\left. - \frac{ 1 }{ \left( \beta - \gamma \mu + (n(l + 1) - d + k)(T - \mu) + d a_{dk} (\min\{\xi_{\mu}^{*},\mu+\mu_{dk}\}-\mu-\mu_{dk}) \right)^{\alpha+i } }\right\} d\mu \right], \\ K_{l6} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g_2(x-T;d-1,\lambda d,- \lambda_{dk}) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha +i) \left[ \sum\limits_{s=0}^{d-2} R_{dks} D_5(d-s-1,\alpha+i,n(l+1),\eta_1) + \frac{1}{(1-a_{dk})^{d-1}} \right. \\ && \times {\int}_{0}^{\eta_1} \mu^j \left\{ \frac{ 1 }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) \right)^{\alpha+i } } \right. \\ &&\left.\left. - \frac{ 1 }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) + d a_{dk} (\min\{\xi_{\mu}^{*},T\}-T) \right)^{\alpha+i } }\right\} d\mu \right], \\ K_{l7} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty} \lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)-d+k} g(x-\mu-\mu_{dk};d-1,\lambda d) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) D_4(d-1,\alpha+i,n(l+1)-d+k,\eta_1), \\ K_{l8}& = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g(x-T;d-1,\lambda d) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) D_5(d-1,\alpha+i,n(l+1),\eta_1), \\ K_{l9} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{nl} g_1(x-\mu;r-1,\lambda r,\lambda n) \ dx d\lambda d\mu \\ &= & \left\{\begin{array}{lll} {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{r-2} R_{s} D_{6}(r-s-1, \alpha+i, nl, \eta_1)\right. \\ \left. + \frac{ 1 }{ (1-\frac{n}{r})^{r-1} } \int\limits_{0}^{\eta_1} \frac{ \mu^j }{ \left( \beta-\gamma \mu + nl(T-\mu) + n (\xi_{\mu}^{*}-\mu) \right)^{\alpha+i } } d\mu \right\}, & \text{if~} r<n \\ {\Gamma}(\alpha+i) D_6(r,\alpha+i,nl,\eta_1), & \text{if~} r=n. \end{array}\right. \end{array} $$

with \(\xi _{6}^{*}(\mu ) = \frac { r (\xi _{\mu }^{*}-\mu ) }{(\beta -\gamma \mu + nl(T-\mu ))}\) and \( R_{s} =(-1)^{s} \frac {n/r}{((n/r)-1)^{s+1}} \),

$$ \begin{array}{@{}rcl@{}} K_{l10} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g_1(x-T;r-1,\lambda r,\lambda n) \ dx d\lambda d\mu \\ &= & \left\{\begin{array}{lll} {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{r-2} R_{s} D_{7}(r-s-1, \alpha+i, n(l+1), \eta_1) \right.\\ \left.+ \frac{ 1 }{ (1-\frac{n}{r})^{r-1} } \int\limits_{0}^{\eta_1} \frac{ \mu^j }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) + n (\xi_T^{*}-T) \right)^{\alpha+i } } d\mu \right\}, & \text{if~} r<n \\ {\Gamma}(\alpha+i) D_7(r,\alpha+i,n(l+1),\eta_1), & \text{if~} r=n \end{array}\right. \end{array} $$

with \(\xi _{7}^{*}(\mu ) = \frac { r (\xi _{T}^{*}-T) }{(\beta -\gamma \mu + n(l+1)(T-\mu ))}\),

$$ \begin{array}{@{}rcl@{}} K_{l11} & = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)-r+k+1} g_1(x - \mu - \mu_{k};r - 1,\lambda r, \lambda_k) \ dx d\lambda d\mu \\ & = & \left\{\begin{array}{lll} {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{r-2} R_{ks} D_{8}(r-s-1, \alpha+i, n(l+1)-r+k+1, \eta_1) \right.\\ \left.\!+ \frac{ 1 }{ (1 - a_k)^{r - 1} } \int\limits_{0}^{\eta_1} \frac{ \mu^j }{ \left( \beta - \gamma \mu + (n(l + 1) - r + k + 1)(T - \mu) + r a_k (\max\{\xi, \mu + \mu_{k}\} - \mu - \mu_k) \right)^{\alpha + i } } d\mu \right\}, & \text{if~} r\!<\!n \\ {\Gamma}(\alpha+i) D_8(r,\alpha+i,n(l+1)-r+k+1,\eta_1), & \text{if~} r = n \end{array}\right. \end{array} $$

with \(\xi _{8}^{*}(\mu ) = \frac { r (\max \limits \{\xi , \mu +\mu _{k}\}-\mu -\mu _{k}) }{(\beta -\gamma \mu + (n(l+1)-r+k+1)(T-\mu ))}\),

$$ \begin{array}{@{}rcl@{}} K_{l12} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g_1(x-T;r-1,\lambda r, \lambda_k) \ dx d\lambda d\mu \\ &= & \left\{\begin{array}{lll} {\Gamma}(\alpha+i) \left\{ \sum\limits_{s=0}^{r-2} R_{ks} D_{9}(r-s-1, \alpha+i, n(l+1), \eta_1) \right.\\ \left.+ \frac{ 1 }{ (1-a_k)^{r-1} } \int\limits_{0}^{\eta_1} \frac{ \mu^j }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) + r a_k (\xi_T^{*}-T) \right)^{\alpha+i } } d\mu \right\}, & \text{if~} r<n \\ {\Gamma}(\alpha+i) D_9(r,\alpha+i,n(l+1),\eta_1), & \text{if~} r=n \end{array}\right. \end{array} $$

with \(\xi _{9}^{*}(\mu ) = \frac { r (\xi _{T}^{*}-T) }{(\beta -\gamma \mu + n(l+1)(T-\mu ))}\),

$$ \begin{array}{@{}rcl@{}} K_{l13} & = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta - \gamma \mu)} q^{n(l+1)-r+k + 1} g_2(x - \mu - \mu_{k};r - 1,\lambda r, - \lambda_k) \ dx d\lambda d\mu \end{array} $$
$$ \begin{array}{@{}rcl@{}} & = & {\Gamma}(\alpha + i) \left[ \sum\limits_{s=0}^{r-2} R_{ks} D_8(r - s - 1,\alpha + i,n(l + 1) - r+k+1,\eta_1) + \frac{1}{(1-a_{k})^{r-1}} \right.\\ && \!\times\! {\int}_{0}^{\eta_1} \mu^j \left\{ \left( \beta-\gamma \mu + (n(l+1)-r+k+1)(T-\mu) \right)^{-\alpha-i } \right.\\ && \left.\left.- \left( \beta-\gamma \mu + (n(l+1)-r+k+1)(T-\mu) + r a_{k} (\min\{\xi_{\mu}^{*},\mu+\mu_{k}\}\right.\right.\right.\\ &&\left.\left.\left.-\mu-\mu_{k}) \right)^{-\alpha-i } \right\} d\mu \right], \\ K_{l14} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g_2(x - T;r - 1,\lambda r,- \lambda_k) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha +i) \left[ \sum\limits_{s=0}^{r-2} R_{ks} D_9(r-s-1,\alpha+i,n(l+1),\eta_1) + \frac{1}{(1-a_{k})^{r-1}} \right. \\ && \times {\int}_{0}^{\eta_1} \mu^j \left\{ \frac{ 1 }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) \right)^{\alpha+i } } \right.\\ &&\left.\left. - \frac{ 1 }{ \left( \beta-\gamma \mu + n(l+1)(T-\mu) + r a_{k} (\min\{\xi_{\mu}^{*},T\}-T) \right)^{\alpha+i } }\right\} d\mu \right], \\ K_{l15} & = & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\!\lambda^{\alpha \!+i - 1} \mu^{j} e^{- \lambda (\beta-\!\gamma \mu)} q^{n(l + 1) - r + k + 1} g(x - \mu - \mu_{k};r - 1,\lambda r)\! \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) D_8(r-1,\alpha+i,n(l+1)-r+k+1,\eta_1) \ \text{and} \\ K_{l16} &= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} {\int}_{\xi_{\mu}^{*}}^{\infty}\lambda^{\alpha +i-1} \mu^{j} e^{- \lambda (\beta-\gamma \mu)} q^{n(l+1)} g(x-T;r-1,\lambda r) \ dx d\lambda d\mu \\ &= & {\Gamma}(\alpha+i) D_9(r-1,\alpha+i,n(l+1),\eta_1). \end{array} $$

Computation of \(E_{\lambda ,\mu }E_{\underline {X}|\lambda ,\mu }(\tau |\lambda ,\mu )\)

$$ \begin{array}{@{}rcl@{}} && E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu) \\ & =& {\int}_{\mu} {\int}_{\lambda} E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu) p(\lambda,\mu) d\lambda d\mu \\ & =&\frac{1}{A}{\int}_{0}^{\eta_1} {\int}_{0}^{\infty} E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu) \lambda^{\alpha-1} e^{- \lambda(\beta-\gamma \mu)} d\lambda d\mu \\ & =& \frac{1}{A}{\int}_{0}^{\eta_1} {\int}_{0}^{\infty} E_{\underline{X}|\lambda,\mu}\left[min\{X_{(r)},T\}|\lambda,\mu\right] \lambda^{\alpha-1} e^{- \lambda(\beta-\gamma \mu)} d\lambda d\mu \\ & =& \frac{1}{A}{\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \left\{ {\int}_{\mu}^{T} y f_{X_{(r)}}(y) dy +T P_{\underline{X}|\lambda,\mu}\left[0 \leq D \leq r-1|\lambda,\mu\right] \right\} \lambda^{\alpha-1} e^{- \lambda(\beta-\gamma \mu)} d\lambda d\mu\\ & =& \frac{1}{A} \sum\limits_{l=0}^{r-1} (-1)^{r-1-l} (n-r+1) \binom{n}{r-1} \binom{r-1}{l} \\ && \times \left[ \frac{1}{n-l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1} \mu e^{-\lambda (\beta- \gamma \mu)} d\lambda d\mu - \frac{T}{n-l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1} e^{-\lambda (\beta+ (n-l)T - (\gamma + n - l) \mu)} d\lambda d\mu \right. \\ && \left. - \frac{1}{(n-l)^2} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-2} e^{-\lambda (\beta+ (n-l)T- (\gamma+ n-l) \mu)} d\lambda d\mu\right.\\ &&\left.+ \frac{1}{(n-l)^2} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-2} e^{-\lambda (\beta- \gamma \mu)} d\lambda d\mu \right] \\ && + \frac{T}{A} \sum\limits_{d=0}^{r-1} \sum\limits_{l=0}^{d} (-1)^{d-l} \binom{n}{d} \binom{d}{l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1} e^{-\lambda (\beta+ (n-l)T- (\gamma+ n-l) \mu)} d\lambda d\mu \end{array} $$

Note that β + (nl)T − (γ + (nl))μ > 0. Hence all the integrals except \( {\int \limits }_{0}^{\eta _{1}} {\int \limits }_{0}^{\infty } \lambda ^{\alpha -2} exp(-\lambda (\beta + (n-l)T- (\gamma + n-l) \mu )) d\lambda d\mu \) and \( {\int \limits }_{0}^{\eta _{1}} {\int \limits }_{0}^{\infty } \lambda ^{\alpha -2} exp(-\lambda \) (βγμ))dλdμ exist for the entire range of hyper parameter α. These two integrals do not exist when 0 < α ≤ 1, so this range is excluded and the Bayes risk has been provided only when α > 1. Hence

$$ \begin{array}{@{}rcl@{}} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu) \!\!& = &\!\! \frac{1}{A} \sum\limits_{l=0}^{r-1} (-1)^{r-1-l} (n-r+1) \binom{n}{r-1} \binom{r-1}{l} \left[ \frac{1}{n-l} {\int}_{0}^{\eta_1} {\Gamma}(\alpha) \frac{\mu }{(\beta- \gamma \mu)^{\alpha}} d\mu \right. \\ && - \frac{T}{n-l} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha)}{(\beta+ (n-l)T- (\gamma+ n-l) \mu)^{\alpha}} d\mu \\ && - \frac{1}{(n-l)^2} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha-1)}{(\beta+ (n-l)T- (\gamma+ n-l) \mu)^{\alpha-1}} d\mu \\ &&\left. + \frac{1}{(n-l)^2} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha-1)}{(\beta- \gamma \mu)^{\alpha-1}} d\mu \right] \\ && + \frac{T}{A} \sum\limits_{d=0}^{r-1} \sum\limits_{l=0}^{d} (-1)^{d-l} \binom{n}{d} \binom{d}{l} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha)}{(\beta+ (n-l)T- (\gamma+ n-l) \mu)^{\alpha}} d\mu. \end{array} $$
(A.2)

Now there are two cases:

Case (1)::

When α≠ 2, Eq. A.2 is given by

$$ \begin{array}{@{}rcl@{}} && E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu} (\tau|\lambda,\mu) = \frac{1}{A} \sum\limits_{l=0}^{r-1} (-1)^{r-l-1} (n-r+1) \binom{n}{r-1} \binom{r-1}{l} \left\{ \frac{\beta ~ {\Gamma}(\alpha -1)}{(n-l)\gamma^2} \right.\\ &&\times \left( \frac{\beta^{\alpha-1}-{(\beta-\gamma \eta_1)}^{\alpha-1}}{\beta^{\alpha-1}{(\beta-\gamma \eta_1)}^{\alpha-1}} \right) - \frac{(\alpha-1) ~ {\Gamma}(\alpha -2)}{(n-l)\gamma^2} \left( \frac{\beta^{\alpha-2}-{(\beta-\gamma \eta_1)}^{\alpha-2}}{\beta^{\alpha-2}{(\beta-\gamma \eta_1)}^{\alpha-2}} \right)\\ && - \frac{T ~ {\Gamma}{(\alpha -1)}}{(n-l) (\gamma+n-l)} \left( \frac{(\beta+(n-l)T)^{\alpha-1}-{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-1}}{(\beta+(n-l)T)^{\alpha-1}{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-1}} \right) \\ && - \frac{ {\Gamma}{(\alpha -2)}}{(n-l)^2 (\gamma+n-l)} \left( \frac{(\beta+(n-l)T)^{\alpha-2}-{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-2}}{(\beta+(n-l)T)^{\alpha-2}{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-2}} \right) \\ &&\left. +\frac{\Gamma{(\alpha -2)}}{(n-l)^2 \gamma} \left( \frac{\beta^{\alpha-2}-{(\beta-\gamma \eta_1)}^{\alpha-2}}{\beta^{\alpha-2}{(\beta-\gamma \eta_1)}^{\alpha-2}} \right) \right\}+ \frac{T}{A} \sum\limits_{d=0}^{r-1} \sum\limits_{l=0}^{d} (-1)^{d-l} \binom{n}{d} \binom{d}{l} \frac{\Gamma{(\alpha -1)}}{(\gamma+n-l)} \\ && \times \left\{ \frac{(\beta+(n-l)T)^{\alpha-1}-{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-1}}{(\beta+(n-l)T)^{\alpha-1}{(\beta+(n-l)T-\eta_1(\gamma+n-l))}^{\alpha-1}} \right\}. \end{array} $$
Case (2)::

When α = 2, simplifying Eq. A.2 further the following expression is obtained:

$$ \begin{array}{@{}rcl@{}} && E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu} (\tau|\lambda,\mu)= \frac{1}{A} \sum\limits_{l=0}^{r-1} (-1)^{r-l-1} (n-r+1) \binom{n}{r-1} \binom{r-1}{l} \left\{ \frac{\eta_1}{ (n-l) \gamma (\beta - \gamma \eta_1)}\right. \\ && - \frac{ln(\beta)-ln(\beta - \gamma \eta_1)}{(n-l) \gamma^2 } - \frac{T}{(n-l)} \left( \frac{\eta_1}{(\beta+(n-l)T){(\beta+(n-l)T-\eta_1(\gamma+n-l))}} \right) \\ &&\left. - \frac{ln(\beta+(n-l)T)-ln(\beta+(n-l)T-\eta_1(\gamma+n-l))}{(n-l)^2 (\gamma+n-l)} + \frac{ln(\beta)-ln(\beta-\gamma \eta_1)}{(n-l)^2 \gamma} \right\} \\ && + \frac{T}{A} \sum\limits_{d=0}^{r-1} \sum\limits_{l=0}^{d} (-1)^{d-l} \binom{n}{d} \binom{d}{l} \left\{\frac{\eta_1}{(\beta+(n-l)T){(\beta+(n-l)T-\eta_1(\gamma+n-l))}} \right\}. \end{array} $$

Computation of \(E_{\lambda ,\mu }E_{\underline {X}|\lambda ,\mu }(D|\lambda ,\mu )\)

$$ \begin{array}{@{}rcl@{}} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu)&= & {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) p(\lambda,\mu) d\lambda d\mu \\ &= & \frac{1}{A} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \left\{ \sum\limits_{d=0}^{r-1} d \binom{n}{d} e^{- \lambda (n-d)(T-\mu)} {\left( 1- e^{- \lambda(T-\mu)} \right)}^d \right. \\ &&\left. + r \sum\limits_{k=r}^{n} \binom{n}{k} e^{- \lambda (n-k)(T-\mu)} {\left( 1- e^{- \lambda(T-\mu)} \right)}^k \right\} \lambda^{\alpha-1} e^{- \lambda(\beta -\gamma \mu)} d\lambda d\mu \\ &= & \frac{1}{A} \sum\limits_{d=0}^{r-1} \sum\limits_{l=0}^{d} (-1)^{d-l} d \binom{n}{d} \binom{d}{l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1}e^{-\lambda \left[\beta+(n-l)T -(\gamma +n-l)\mu\right]} d\lambda d\mu \\ && + \frac{r}{A} \sum\limits_{k=r}^{n} \sum\limits_{l=0}^{k} (\!-1)^{k-l} \binom{n}{k} \binom{k}{l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1} e^{-\lambda \left[\beta+(n-l)T -(\gamma +n-l)\mu\right]} d\lambda d\mu\\ && = \frac{1}{A} \sum\limits_{d=0}^{n} \sum\limits_{l=0}^{d} (- 1)^{d-l} m \binom{n}{d} \binom{d}{l} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha - 1}e^{-\!\lambda \left[\beta+(n\!-l)T \!-(\gamma +n-l)\mu\right]} d\lambda d\mu \end{array} $$

where \(m=\min \limits \{d,r\}\). Hence

$$ \begin{array}{@{}rcl@{}} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) & =&\frac{1}{A} \sum\limits_{d=0}^{n} \sum\limits_{l=0}^{d} (-1)^{d-l} m \binom{n}{d} \binom{d}{l} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha)}{(\beta+ (n-l)T- (\gamma+ n-l) \mu)^{\alpha}} d\mu \\ && =\frac{1}{A} \sum\limits_{d=0}^{n} \sum\limits_{l=0}^{d} (-1)^{d-l} m \binom{n}{d} \binom{d}{l} \frac{\Gamma(\alpha-1)}{\gamma +n-l} \\ &&\times \left\{ \frac{(\beta+(n-l)T)^{\alpha-1}-(\beta+(n-l)T-(\gamma +n-l)\eta_1)^{\alpha-1}}{(\beta+(n-l)T)^{\alpha-1} (\beta+(n-l)T-(\gamma +n-l)\eta_1 )^{\alpha-1}}\right\} \end{array} $$

Computation of E λ, μ(e nλ(Tμ))

$$ \begin{array}{@{}rcl@{}} E_{\lambda,\mu}(e^{-n \lambda (T-\mu)}) &=& {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} e^{-n \lambda(T-\mu)} p(\lambda,\mu) d\lambda d\mu \\ & =& \frac{1}{A}{\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha-1} e^{- \lambda(\beta+nT- (n+\gamma) \mu ) } d\lambda d\mu \\ & =& \frac{1}{A}{\int}_{0}^{\eta_1} \frac{\Gamma(\alpha)}{(\beta +nT - (n+\gamma) \mu )^{\alpha}} d\mu \\ & = &\frac{1}{A} \frac{\Gamma{(\alpha - 1)}}{n + \gamma} \left\{ \frac{(\beta + nT)^{\alpha-1} - {(\beta+nT-(n+\gamma) \eta_1)}^{\alpha-1}}{(\beta + nT)^{\alpha-1}{(\beta+nT-(n+\gamma) \eta_1)}^{\alpha-1}} \right\} \end{array} $$

Computation of ∑1≤i+j≤ 2 C ij E λ, μ(λ i μ j e nλ(Tμ))

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}(\lambda^i \mu^j e^{-n \lambda (T-\mu)}) \\ &=& \sum\limits_{1 \leq i+j \leq 2} C_{ij} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^i \mu^j e^{-n \lambda (T-\mu)} p(\lambda,\mu) d\lambda d\mu \\ & =& \frac{1}{A}\sum\limits_{1 \leq i+j \leq 2} C_{ij} {\int}_{0}^{\eta_1} {\int}_{0}^{\infty} \lambda^{\alpha+i-1} \mu^j e^{- \lambda (\beta+nT- (n+\gamma) \mu ) } d\lambda d\mu\\ & = &\frac{1}{A}\sum\limits_{1 \leq i+j \leq 2} C_{ij} {\int}_{0}^{\eta_1} \frac{\Gamma(\alpha+i) }{(\beta+nT-(n+\gamma )\mu)^{\alpha+i}} \mu^j d\mu \\ & =& \frac{1}{A} \sum\limits_{1 \leq i+j \leq 2} C_{ij} \frac{\Gamma(\alpha+i)}{(\beta+nT)^{\alpha+i}} {\int}_{0}^{\frac{(n+\gamma ) \eta_1}{\beta+nT}} {\left( \frac{\beta+nT}{(n+\gamma )}\right)}^{j+1} \frac{u^j}{(1-u)^{\alpha+i}} du \\ & =& \frac{1}{A} \sum\limits_{1 \leq i+j \leq 2} C_{ij} \frac{\Gamma(\alpha+i)}{(n+\gamma )^{j+1}(\beta+nT)^{\alpha+i-j-1}} {\int}_{0}^{\eta*} \frac{u^j}{(1-u)^{\alpha+i}} du \\ &=& C_{01} \frac{\Gamma(\alpha)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-2}} {\int}_{0}^{\eta*} \frac{u}{(1-u)^{\alpha}} du \\&&+ C_{10} \frac{\Gamma(\alpha+1)}{A (n+\gamma ) (\beta+nT)^{\alpha}} {\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha+1}} du \\ && + C_{11} \frac{\Gamma(\alpha+1)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-1}} {\int}_{0}^{\eta*} \frac{u}{(1-u)^{\alpha+1}} du \\&&+C_{20} \frac{\Gamma(\alpha+2)}{A (n+\gamma ) (\beta+nT)^{\alpha+1}}\times {\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha+2}} du \\&&+ C_{02} \frac{\Gamma(\alpha)}{A (n+\gamma )^3 (\beta+nT)^{\alpha-3}} {\int}_{0}^{\eta*} \frac{u^2}{(1-u)^{\alpha}} du, \end{array} $$

where \( \eta *= \frac {(n+\gamma )\eta _{1}}{\beta +nT} \) and since \({\int \limits }_{0}^{\eta *} \frac {u}{(1-u)^{p}} du = -{\int \limits }_{0}^{\eta *} \frac {1}{(1-u)^{p-1}} du+{\int \limits }_{0}^{\eta *} \frac {1}{(1-u)^{p}} du\) and \({\int \limits }_{0}^{\eta *} \frac {u^{2}}{(1-u)^{p}} du ={\int \limits }_{0}^{\eta *} \frac {1}{(1-u)^{p}} du+{\int \limits }_{0}^{\eta *} \frac {1}{(1-u)^{p-2}} du- 2{\int \limits }_{0}^{\eta *} \frac {1}{(1-u)^{p-1}} du\). Therefore,

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}(\lambda^i \mu^j e^{-n \lambda (T-\mu)}) \\ && = C_{01} \frac{\Gamma(\alpha)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-2}} \left\{ -{\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha-1}} du+{\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha}} du \right\} \\ && + C_{10} \frac{\Gamma(\alpha + 1)}{A (n + \gamma ) (\beta + nT)^{\alpha}} {\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha+1}} du + C_{11} \frac{\Gamma(\alpha+1)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-1}} \\ && \times \left\{ - {\int}_{0}^{\eta*} \frac{1}{(1 - u)^{\alpha}} du +{\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha+1}} du \right\} +C_{20} \frac{\Gamma(\alpha+2)}{A (n+\gamma ) (\beta+nT)^{\alpha+1}} \\ && \times {\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha+2}} du + C_{02} \frac{\Gamma(\alpha)}{A (n+\gamma )^3 (\beta+nT)^{\alpha-3}} \times \left\{ {\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha}} du \right. \\ &&\left. +{\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha-2}} du - 2{\int}_{0}^{\eta*} \frac{1}{(1-u)^{\alpha-1}} du \right\}. \end{array} $$

Case (1): Thus when α∉{2,3}, we get,

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}(\lambda^i \mu^j e^{-n \lambda (T-\mu)}) \\ && = C_{01} \frac{\Gamma(\alpha)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-2}} \left\{ -\frac{1}{\alpha-2} \left( \frac{1}{(1-\eta^{*})^{\alpha-2}} -1 \right) + \frac{1}{\alpha-1} \left( \frac{1}{(1-\eta^{*})^{\alpha-1}} - 1 \right) \right\}\\ &&+ C_{10} \frac{\Gamma(\alpha+1)}{A (n+\gamma ) (\beta+nT)^{\alpha}} \frac{1}{\alpha} \left( \frac{1}{(1-\eta^{*})^{\alpha}} -1 \right) + C_{11} \frac{\Gamma(\alpha+1)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-1}} \\ &&\times \left\{ - \frac{1}{\alpha-1} \left( \frac{1}{(1-\eta^{*})^{\alpha-1}} -1 \right) + \frac{1}{\alpha} \left( \frac{1}{(1-\eta^{*})^{\alpha}} -1 \right) \right\} +C_{20} \frac{\Gamma(\alpha+2)}{A (n+\gamma ) (\beta+nT)^{\alpha+1}} \\ &&\times \frac{1}{\alpha+1} \left( \frac{1}{(1-\eta^{*})^{\alpha+1}} -1 \right) + C_{02} \frac{\Gamma(\alpha)}{A (n+\gamma )^3 (\beta+nT)^{\alpha-3}} \left\{ \frac{1}{\alpha-1} \left( \frac{1}{(1-\eta^{*})^{\alpha-1}} -1 \right) \right. \\ &&\left. + \frac{1}{\alpha-3} \left( \frac{1}{(1-\eta^{*})^{\alpha-3}} -1 \right) - 2 \frac{1}{\alpha-2} \left( \frac{1}{(1-\eta^{*})^{\alpha-2}} -1 \right) \right\} \\ & =& \frac{\Gamma(\alpha-1)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-3}} \left( \frac{C_{01}}{\beta+nT}- \frac{C_{11} \alpha}{(\beta+nT)^2} + \frac{C_{02}}{(n+\gamma )}\right) \left( \frac{1}{(1-\eta^{*})^{\alpha-1}}-1 \right) \\ && - \frac{(\alpha-1){\Gamma}(\alpha-2)}{A (n+\gamma )^2 (\beta+nT)^{\alpha-3}} \left( \frac{C_{01}}{\beta+nT}- \frac{2C_{02}}{(n+\gamma )}\right) \left( \frac{1}{(1-\eta^{*})^{\alpha-2}}-1 \right) \\ && + \frac{C_{02} {\Gamma}(\alpha) }{A (\alpha-3)(n+\gamma )^3 (\beta+nT)^{\alpha-3}} \left( \frac{1}{(1-\eta^{*})^{\alpha-3}}-1 \right) + \frac{\Gamma(\alpha)}{A (n+\gamma ) (\beta+nT)^{\alpha-1}} \\ &&\times \left( \frac{C_{10}}{\beta+nT}+ \frac{C_{11}}{(n+\gamma )}\right) \left( \frac{1}{(1-\eta^{*})^{\alpha}}-1 \right) + \frac{C_{20}{\Gamma}(\alpha+1)}{A (n+\gamma ) (\beta+nT)^{\alpha+1}} \left( \frac{1}{(1-\eta^{*})^{\alpha+1}}-1 \right), \end{array} $$

Case (2): when α = 2.

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}(\lambda^i \mu^j e^{-n \lambda (T-\mu)}) \\ & =& C_{01} \frac{\Gamma(2)}{A (n+\gamma )^2 } \left\{ ln(1- \eta^{*})+ \left( \frac{1}{(1-\eta^{*})} -1 \right) \right\} + C_{10} \frac{\Gamma(3)}{2 A (n+\gamma ) (\beta+nT)^{2}} \left( \frac{1}{(1-\eta^{*})^2} -1 \right) \\ && + C_{11} \frac{\Gamma(3)}{A (n+\gamma )^2 (\beta+nT)} \left\{ - \left( \frac{1}{(1-\eta^{*})} -1 \right) +\frac{1}{2} \left( \frac{1}{(1-\eta^{*})^2} -1 \right) \right\} \\ && + C_{20} \frac{\Gamma(4)}{A (n+\gamma ) (\beta+nT)^3} \frac{1}{3} \left( \frac{1}{(1-\eta^{*})^3} -1 \right) + C_{02} \frac{\Gamma(2)}{A (n+\gamma )^3 (\beta+nT)^{-1}} \\ && \times \left\{ \left( \frac{1}{(1-\eta^{*})} -1 \right) +\eta^{*} +2 ln(1-\eta^{*}) \right\} \\ & =& \frac{1}{A (n+\gamma )^2 } \left( C_{01}- \frac{2C_{11}}{\beta+nT}+\frac{C_{02} (\beta+nT)}{(n+\gamma )} \right) \left( \frac{\eta^{*}}{1-\eta^{*}} \right) + \frac{1}{A (n+\gamma )^2 } \left( C_{01} + \frac{2C_{02}(\beta+nT)}{(n+\gamma )} \right) \\ && \times ln(1-\eta^{*}) + \frac{1}{A (n+\gamma ) (\beta+nT)} \left( \frac{C_{10}}{\beta+nT} + \frac{C_{11}}{(n+\gamma )} \right) \left( \frac{1}{(1-\eta^{*})^2} -1 \right) \\ && + \frac{2C_{20}}{A (n+\gamma ) (\beta+nT)^3} \left( \frac{1}{(1-\eta^{*})^3} -1 \right) + \frac{C_{02} (\beta+nT)}{A (n+\gamma )^3} \eta^{*}, \end{array} $$

Case (3): and when α = 3.

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{1 \leq i+j \leq 2} C_{ij} E_{\lambda,\mu}(\lambda^i \mu^j e^{-n \lambda (T-mu)}) \\ & =& C_{01} \frac{\Gamma(3)}{A (n+\gamma )^2 (\beta+nT)} \left\{ - \left( \frac{1}{(1-\eta^{*})} -1 \right) + \frac{1}{2} \left( \frac{1}{(1-\eta^{*})^{2}} -1 \right) \right\} \\ && + C_{10} \frac{\Gamma(4)}{A (n+\gamma ) (\beta+nT)^{3}} \frac{1}{3} \left( \frac{1}{(1-\eta^{*})^{3}} -1 \right) + C_{11} \frac{\Gamma(4)}{A (n+\gamma )^2 (\beta+nT)^{2}} \\ &&\times \left\{ - \frac{1}{2} \left( \frac{1}{(1-\eta^{*})^{2}} -1 \right)+ \frac{1}{3} \left( \frac{1}{(1-\eta^{*})^{3}} -1 \right) \right\} +C_{20} \frac{\Gamma(5)}{A (n+\gamma ) (\beta+nT)^{4}} \\ && \times \frac{1}{4} \left( \frac{1}{(1-\eta^{*})^{4}} -1 \right) + C_{02} \frac{\Gamma(3)}{A (n+\gamma )^3 } \left\{ \frac{1}{2} \left( \frac{1}{(1-\eta^{*})^{2}} -1 \right)- ln(1-\eta^{*}) - 2 \left( \frac{1}{(1-\eta^{*})} -1 \right) \right\} \\ & = &\frac{1}{A (n+\gamma )^2 } \left( \frac{C_{01}}{\beta+nT} -\frac{3C_{11}}{(\beta+nT)^2}+\frac{C_{02}}{(n+\gamma )} \right)\left( \frac{1}{(1-\eta^{*})^{2}} -1 \right) - \frac{1}{A (n+\gamma )^2 } \left( \frac{2C_{01}}{\beta+nT} \right. \\ &&\left. +\frac{C_{02}}{(n+\gamma )} \right) \left( \frac{\eta^{*}}{1-\eta^{*}}\right) +\frac{2}{A (n+\gamma ) (\beta+nT)^2 } \left( \frac{C_{10}}{\beta+nT} -\frac{C_{11}}{(n+\gamma )} \right)\left( \frac{1}{(1-\eta^{*})^{3}} -1 \right) \\ && + \frac{6C_{20}}{A (n+\gamma ) (\beta+nT)^4} \left( \frac{1}{(1-\eta^{*})^4} -1 \right) - \frac{2C_{02}}{A (n+\gamma )^3} ln(1-\eta^{*}). \end{array} $$

Appendix B: Proof of Theorem 4.1

Suppose (0,0,0,0) denotes the sampling plan which accepts the lot without sampling (n = 0 or 1). It is easy to verify from the Eq. A.1 that the corresponding Bayes risk

$$ \begin{array}{@{}rcl@{}} r(0,0,0,0)= \sum\limits_{0 \leq i+j \leq 2}C_{ij} E_{\lambda,\mu}\left( \lambda^{i} \mu^{j}\right), \end{array} $$

for expressions of the required moments, see Lam (1990). Similarly if (0,0,0, \(\infty )\) denotes the sampling plan which rejects the lot without sampling (n = 0), then the corresponding Bayes risk \(r(0,0,0,\infty )\) will be Cr. Obviously the Bayes risk corresponding to the optimal Bayesian sampling plan satisfies the inequality

$$ \begin{array}{@{}rcl@{}} && r(n_{0},r_{0},T_{0},\xi_{0}) \leq \min\{ r(0,0,0,0),r(0,0,0,\infty)\} \\ && \Rightarrow r(n_{0},r_{0},T_{0},\xi_{0}) \leq \min\left\{\sum\limits_{0 \leq i+j \leq 2}C_{ij} E_{\lambda,\mu}\left( \lambda^{i} \mu^{j}\right),C_{r}\right\} \end{array} $$
(A.3)

Also, by Eq. A.1

$$ \begin{array}{@{}rcl@{}} r(n_{0},r_{0},T_{0},\xi_{0}) &= & n_{0}(C_{s}-r_{s})+C_{T} E_{\lambda,\mu}E_{\underline{}|\lambda,\mu}(\tau|\lambda,\mu)+ r_{s} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) + C_{r}\\ && + E_{\lambda,\mu} E_{\underline{X}|\lambda,\mu} \left[\delta(\underline{X})\left( \sum\limits_{0 \leq i+j \leq 2}{C_{ij} \lambda^{i} \mu^{j}}-C_{r}\right)\right] \\ &= & n_{0}(C_{s}-r_{s})+C_{T} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(\tau|\lambda,\mu)+ r_{s} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}(D|\lambda,\mu) \\ && + C_{r} E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}\left( 1- \delta(\underline{X}) \right) + E_{\lambda,\mu}E_{\underline{X}|\lambda,\mu}\left[\delta(\underline{X})\left( \sum\limits_{0 \leq i+j \leq 2}{C_{ij} \lambda^{i} \mu^{j}}\right)\right] \end{array} $$

In the right hand side of above equation, leaving out the first term, all others are positive terms. Therefore

$$ \begin{array}{@{}rcl@{}} r(n_{0},r_{0},T_{0},\xi_{0}) \geq n_{0}(C_{s}-r_{s}). \end{array} $$
(A.4)

Hence Eqs. A.3 and A.4 together imply the first result and of course we have 1 ≤ r0n0.

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Prajapat, K., Koley, A., Mitra, S. et al. An Optimal Bayesian Sampling Plan for Two-Parameter Exponential Distribution Under Type-I Hybrid Censoring. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00263-2

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Keywords

  • Exponential distribution
  • Type-I hybrid censoring
  • Bayesian sampling plan
  • Bayes risk
  • conjugate priors
  • optimal sampling plan.

AMS (2000) subject classification

  • Primary 62F10
  • 62H12
  • 65D30