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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Appendices
Appendix A: Proof of Theorem 3.2
Equation A.1 further can be simplified as follows
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, β − γμ + (2l + 1)(T − μ) > 0 and β − γμ + 2(l + 1)(T − μ) > 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 β − γμ + 2l(T − μ) + 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/(β − γμ + 2l(T − μ)) 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 ⇔ 2d − k − n > (<)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 \(E_{\lambda ,\mu }E_{\underline {X}|\lambda ,\mu }(\tau |\lambda ,\mu )\)
Note that β + (n − l)T − (γ + (n − l))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 \( {\int \limits }_{0}^{\eta _{1}} {\int \limits }_{0}^{\infty } \lambda ^{\alpha -2} exp(-\lambda \) (β − γμ))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
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 \(E_{\lambda ,\mu }E_{\underline {X}|\lambda ,\mu }(D|\lambda ,\mu )\)
where \(m=\min \limits \{d,r\}\). Hence
Computation of E λ, μ(e −nλ(T−μ))
Computation of ∑1≤i+j≤ 2 C ij E λ, μ(λ i μ j e −nλ(T−μ))
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,
Case (1): Thus when α∉{2,3}, we get,
Case (2): when α = 2.
Case (3): and when α = 3.
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
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
Also, by Eq. A.1
In the right hand side of above equation, leaving out the first term, all others are positive terms. Therefore
Hence Eqs. A.3 and A.4 together imply the first result and of course we have 1 ≤ r0 ≤ n0.
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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 85, 512–539 (2023). https://doi.org/10.1007/s13171-021-00263-2
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DOI: https://doi.org/10.1007/s13171-021-00263-2
Keywords
- Exponential distribution
- Type-I hybrid censoring
- Bayesian sampling plan
- Bayes risk
- conjugate priors
- optimal sampling plan.