Point and interval estimation of quantiles of several exponential populations with a common location under progressive censoring scheme

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Abstract

We consider the problems of point, and interval estimation of the $p\textrm{th}$ quantile of the first population when progressive type-II censored samples are available from several exponential populations with a common location, and different scale parameters. First, in the case of point estimation, we derive the maximum likelihood estimator, a modification to it and the uniformly minimum variance unbiased estimator (UMVUE) of the quantile. An estimator dominating the UMVUE is derived. Further, a class of affine equivariant estimators is derived, and an inadmissibility result is proved. Consequently, improved estimators dominating the UMVUE are derived. In the case of interval estimation, several confidence intervals, such as generalized confidence interval, bootstrap confidence interval, and the highest posterior density confidence interval, are obtained for the quantile. The point estimators are compared through the risk values, whereas the interval estimators are compared through coverage probability and average length using a simulation study numerically. The conclusions regarding their performances have been made based on our simulation study. Finally, a real-life data set has been considered for illustrative purposes.

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Acknowledgements

The authors would like to sincerely thank the two anonymous reviewers, an associate editor and the editor in chief for their helpful comments and suggestions which have helped significantly to improve the manuscript.

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Department of Mathematics, National Institute of Technology Rourkela, Rourkela, 769008, India
Habiba Khatun & Manas Ranjan Tripathy

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Habiba Khatun
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Appendix

${\textbf{Derivation}~\textbf{of}~\textbf{joint}~\textbf{distribution}~\textbf{of}~U_{i}'s;~ i=1,2,\ldots ,k}$

Denote $X_{ij:m_{i}:n_{i}} = X_{ij},$ $W_{i}= \frac{1}{n_{i}}\sum _{i=1}^{m_{i}}(r_{ij}+1)(X_{ij}-X_{i1}),$ and $U_{i}= W_{i}+(X_{i1}-Z);$ $i=1,2,\ldots ,k.$ Now

$$\begin{aligned}{} & {} P\Big (U_{1}>u_{1},U_{2}>u_{2},\ldots ,U_{k}>u_{k}\Big )\nonumber \\{} & {} \quad = P\Big (W_{1}+(X_{11}-Z)>u_{1},\ldots ,W_{k}+(X_{k1}-Z)>u_{k}\Big )\nonumber \\{} & {} \quad =\sum _{l=1}^{k}P\Big (W_{1}+(X_{11}-Z)>u_{1},\ldots ,W_{k}+(X_{k1}-Z)>u_{k}, X_{l1}= \min _{\begin{array}{c} 1\le i \le k \end{array}}X_{i1}\Big )\nonumber \\{} & {} \quad =\sum _{\begin{array}{c} l=1 \\ l \ne i \end{array}}^{k}P\Big (W_{1}+(X_{11}-Z)>u_{1},\ldots ,W_{k}+(X_{k1}-Z)>u_{k}| X_{i1}>X_{l1} \Big ) P\Big (X_{i1}> X_{l1}\Big )\nonumber \\{} & {} \quad =\sum _{\begin{array}{c} l=1 \\ l \ne i \end{array}}^{k}P\Big (W_{i}+(X_{i1}-X_{l1})>u_{i}, W_{l}>u_{l}| X_{i1}>X_{l1}\Big ) P\Big (X_{i1}> X_{l1}\Big )\nonumber \\{} & {} \quad =\sum _{\begin{array}{c} l=1 \\ l \ne i \end{array}}^{k}P\Big (W_{i}+(X_{i1}-X_{l1})>u_{i}| X_{i1}>X_{l1}\Big ) P\Big ( W_{l}>u_{l}\Big ) P\Big (X_{i1}> X_{l1} \Big ) \end{aligned}$$

(5.4)

Now, for $l=1,2,\ldots ,k$

$$\begin{aligned}{} & {} P\left( X_{i1}> X_{l1}~ \forall ~ i=1, \ldots ,k~(\ne l))=P(X_{i1}-\mu > X_{l1}-\mu ~ \forall ~ i=1, \ldots , k~(\ne l)\right) \nonumber \\{} & {} \quad = \int _{0}^{\infty }\left[ \prod _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\int _{x_{l1}}^{\infty }\frac{n_{i}}{\sigma _{i}}\exp \left( -\frac{n_{i}}{\sigma _{i}}x_{i1}\right) dx_{i1}\right] \frac{n_{l}}{\sigma _{l}}\exp \left( -\frac{n_{l}}{\sigma _{l}}x_{l1}\right) dx_{l1}\nonumber \\{} & {} \quad =\int _{0}^{\infty }\left[ \prod _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\exp \left( -\frac{n_{i}}{\sigma _{i}}x_{l1}\right) \right] \frac{n_{l}}{\sigma _{l}}\exp \left( -\frac{n_{l}}{\sigma _{l}}x_{l1}\right) dx_{l1}\nonumber \\{} & {} \quad =\int _{0}^{\infty }\frac{n_{l}}{\sigma _{l}} \exp \left\{ -\left( \sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}+\frac{n_{l}}{\sigma _{l}}\right) x_{l1}\right\} dx_{l1}\nonumber \\{} & {} \quad =\int _{0}^{\infty }\frac{n_{l}}{\sigma _{l}} \exp \left\{ -\left( \sum _{i=1}^{k}\frac{n_{i}}{\sigma _{i}}\right) x_{l1}\right\} dx_{l1}\nonumber \\{} & {} \quad =\frac{n_{l}}{\sigma _{l}}\left( \sum _{i=1}^{k}\frac{n_{i}}{\sigma _{i}}\right) ^{-1} =\frac{n_{l}}{\sigma _{l}}\sigma ^{-1} \end{aligned}$$

(5.5)

Next consider

$$\begin{aligned}{} & {} P\left( X_{i1}-X_{l1}>u_{i}| X_{i1}>X_{l1}~ \forall ~ i~ (\ne l)\right) =\frac{ P\left( X_{i1}-X_{l1}>u_{i}~ \forall ~ i (\ne l)\right) }{P\left( X_{i1} >X_{l1} ~\forall ~ i~(\ne l)\right) }\nonumber \\{} & {} \quad =\left[ \frac{n_{l}}{\sigma _{l}}\sigma ^{-1}\right] ^{-1}\int _{0}^{\infty }\left[ \prod _{\begin{array}{c} i=1\\ (\ne l) \end{array}}^{k}\int _{u_{i}+x_{l1}}^{\infty }\frac{n_{i}}{\sigma _{i}}\exp \left( -\frac{n_{i}}{\sigma _{i}}x_{i1}\right) dx_{i1}\right] \nonumber \\{} & {} \qquad \times \frac{n_{l}}{\sigma _{l}}\exp \left( -\frac{n_{l}}{\sigma _{l}}x_{l1}\right) dx_{l1}\nonumber \\{} & {} \quad =\sigma \int _{0}^{\infty }\left[ \prod _{\begin{array}{c} i=1\\ (\ne l) \end{array}}^{k}\exp \left( -\frac{n_{i}}{\sigma _{i}}(u_{i}+x_{l1})\right) \right] \exp \left( -\frac{n_{l}}{\sigma _{l}}x_{l1}\right) dx_{l1}\nonumber \\{} & {} \quad =\sigma \exp \left( -\sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}u_{i}\right) \int _{0}^{\infty }\exp \left\{ -\left( \sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}+\frac{n_{l}}{\sigma _{l}}\right) x_{l1}\right\} dx_{l1}\nonumber \\{} & {} \quad =\sigma \exp \left( -\sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}u_{i}\right) \int _{0}^{\infty }\exp \left\{ -\left( \sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}+\frac{n_{l}}{\sigma _{l}}\right) x_{l1}\right\} dx_{l1}\nonumber \\{} & {} \quad =\sigma \exp \left( -\sum _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k}\frac{n_{i}}{\sigma _{i}}u_{i}\right) \left( \sum _{i=1}^{k}\frac{n_{i}}{\sigma _{i}}\right) ^{-1}\nonumber \\{} & {} \quad =\prod _{\begin{array}{c} i=1\\ (\ne l) \end{array}}^{k}\exp \left( -\frac{n_{i}}{\sigma _{i}}u_{i}\right) \end{aligned}$$

(5.6)

This implies, the random variables $(X_{i1}-Z)$ ($\forall ~ i=1,2,\ldots , k (\ne l)$) given $X_{il}>X_{l1}$ are independently distributed as $Gamma(1, n_{i}/\sigma _{i}).$ Further, it is known that $\sum _{j=1}^{m_{i}}(r_{ij}+1)(X_{ij}-X_{i1})$ follows $Gamma(m_{i}-1, 1/\sigma _{i})$ which implies $W_{i}$ follows $Gamma(m_{i}-1, n_{i}/\sigma _{i}).$

The random variables $W_{i}$ and $(X_{i1}-Z)$ $\forall ~ i=1,2,\ldots ,k~ (\ne l)$ are independent. Therefore, $W_{i}+(X_{i1}-Z)$ $\forall ~ i=1,2,\ldots ,k ~(\ne l)$ independently distributed as $Gamma(m_{i}, n_{i}/\sigma _{i})$ conditional on $X_{il}>X_{l1}.$

Utilizing the equations (5.4)–(5.6) we obtain the joint probability density function of $(U_1, U_2, \ldots , U_k)$ as

$$\begin{aligned} f(u_1,u_2,\ldots ,u_k)= & {} \sum _{l=1}^{k}\left[ \frac{n_{l}}{\sigma _{l}}\sigma ^{-1}\left\{ \prod _{\begin{array}{c} i=1\\ (\ne l) \end{array}}^{k}\frac{\left( \frac{n_{i}}{\sigma _{i}}\right) ^{m_{i}}\exp \left( -\frac{n_{i}}{\sigma _{i}}u_{i}\right) u_{i}^{m_{i}-1}}{\Gamma (m_{i})}\right\} \right. \\{} & {} \left. \frac{\left( \frac{n_{l}}{\sigma _{l}}\right) ^{m_{l}-1}\exp \left( -\frac{n_{l}}{\sigma _{l}}u_{l}\right) u_{l}^{m_{l}-2}}{\Gamma (m_{l}-1)}\right] \nonumber \\= & {} \frac{\prod _{i=1}^{k}n_{i}^{m_{i}}}{\sigma \prod _{i=1}^{k}\sigma _{i}^{m_{i}}}\left[ \sum _{l=1}^{k}\left( \prod _{\begin{array}{c} i=1 \\ i \ne l \end{array}}^{k} \frac{u_{i}^{m_{i}-1}}{\Gamma (m_{i})}\right) \frac{u_{l}^{m_{l}-2}}{\Gamma (m_{l}-1)}\right] \exp \left( -\sum _{i=1}^{k}\frac{n_{i}u_{i}}{\sigma _{i}}\right) . \end{aligned}$$

Proof of Theorem 2.1

It may be noted that the statistics Z and $(U_1, U_2, \ldots , U_k)$ are jointly sufficient using Fisher-Neyman factorization theorem. Further the statistics Z and $(U_1, U_2, \ldots , U_k)$ are independent using Basu’s theorem. To show that the statistics are complete, one need to prove that when $E_{\mu , \sigma _{1}, \sigma _{2}, \ldots , \sigma _{k}}H(Z, U_1, U_2, U_k)=0$ the it implies $H(Z, U_1, U_2, \ldots , U_k)=0$ almost everywhere, for all values of the parameters. But,

$$\begin{aligned} E_{\mu ,\sigma _{1},\sigma _{2}, \ldots , \sigma _{k}}H(Z, U_1, U_2, \ldots , U_k)= & {} \int _{0}^{\infty }\int _{0}^{\infty }\ldots \int _{0}^{\infty } \int _{\mu }^{\infty } H(z, u_{1}, u_{2}, \ldots , u_k) \sigma \nonumber \\{} & {} \exp \left( -\sigma (z-\mu )\right) \nonumber \\{} & {} \times f(u_{1}, u_{2}, \ldots , u_k)~ dz du_{1} du_{2}\ldots du_{k}=0.\nonumber \\ \end{aligned}$$

(5.7)

Differentiating the above equation (5.7) with respect to $\mu ,$ under the integral sign, one gets

$$\begin{aligned}{} & {} \int _{0}^{\infty }\ldots \int _{0}^{\infty } \int _{0}^{\infty } H(\mu , u_{1}, u_{2}, \ldots , u_k) f(Z, u_{1}, u_{2}, \ldots , u_k) du_{1}du_{2} \ldots du_k= 0,\\{} & {} \quad \text{ almost } \text{ everywhere }. \end{aligned}$$

Next, using the arguments given in Theorem 2.5.1 of Razmpour (1982), it is easy to show that the family of distributions generated by the statistics $(Z, U_1, U_2, \ldots , U_k)$ is complete. Thus the joint statistics $(Z, U_1, U_2, \ldots , U_k)$ is complete and sufficient for $(\mu , \sigma _{1}, \sigma _{2}, \ldots , \sigma _{k}).$

To derive the UMVUEs of $\mu $ and $\sigma _{i}$s, we proceed as follows. Consider

(5.8)

Now substituting this in $E(\hat{\mu }_{MV})$ one gets the UMVUE of $\mu .$ In a similar manner one can derive the UMVUE of $\sigma _{i};$ $i=1,2,\ldots ,k.$ $\square $

${\textbf{Derivation}~\textbf{of}~\textbf{first}~\textbf{and}~\textbf{second}~\textbf{moments}~\textbf{of}~ U_{j}{} \textbf{s}:}$

The marginal density of $U_{j}$ is given by

(5.9)

The first moment of $U_{j}$ is obtained as

$$\begin{aligned} E(U_{j})= & {} \int _{0}^{\infty }u_{j} f_{U_{j}}(u_{j}) du_{j}\nonumber \\= & {} \int _{0}^{\infty } \frac{n_{j}^{m_{j}}}{\sigma \sigma _{j}^{m_{j}}}\left[ \frac{u_{j}^{m_{j}}}{\Gamma (m_{j})}\left( \sigma -\frac{n_{j}}{\sigma _{j}}\right) +\frac{u_{j}^{m_{j}-1}}{\Gamma (m_{j}-1)}\right] \exp \left( -\frac{n_{j}u_{j}}{\sigma _{j}}\right) du_{j}\nonumber \\= & {} \frac{n_{j}^{m_{j}}}{\sigma \sigma _{j}^{m_{j}}} \left[ \frac{\Gamma (m_{j}+1)\sigma _{j}^{m_{j}+1}}{\Gamma (m_{j})n_{j}^{m_{j}+1}}\left( \sigma -\frac{n_{j}}{\sigma _{j}}\right) +\frac{\Gamma (m_{j})\sigma _{j}^{m_{j}}}{\Gamma (m_{j}-1)n_{j}^{m_{j}}}\right] \nonumber \\= & {} \frac{m_{j}\sigma _{j}}{n_{j}}-\frac{1}{\sigma } \end{aligned}$$

(5.10)

In a similar manner, the second moment of $U_{j}$ is obtained as

$$\begin{aligned} E(U_{j}^2)= & {} \int _{0}^{\infty }u_{j}^2 f_{U_{j}}(u_{j}) du_{j}\nonumber \\= & {} \int _{0}^{\infty } \frac{n_{j}^{m_{j}}}{\sigma \sigma _{j}^{m_{j}}}\left[ \frac{u_{j}^{m_{j}+1}}{\Gamma (m_{j})}\left( \sigma -\frac{n_{j}}{\sigma _{j}}\right) +\frac{u_{j}^{m_{j}}}{\Gamma (m_{j}-1)}\right] \exp \left( -\frac{n_{j}u_{j}}{\sigma _{j}}\right) du_{j}\nonumber \\= & {} \frac{n_{j}^{m_{j}}}{\sigma \sigma _{j}^{m_{j}}} \left[ \frac{\Gamma (m_{j}+2)\sigma _{j}^{m_{j}+1}}{\Gamma (m_{j})n_{j}^{m_{j}+1}}\left( \sigma -\frac{n_{j}}{\sigma _{j}}\right) +\frac{\Gamma (m_{j}+1)\sigma _{j}^{m_{j}}}{\Gamma (m_{j}-1)n_{j}^{m_{j}}}\right] \nonumber \\= & {} \frac{\sigma _{j}}{n_{j}}\left( m_{j}(m_{j}+1)\frac{\sigma _{j}}{n_{j}}-\frac{2 m_{j}}{\sigma }\right) . \end{aligned}$$

(5.11)

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Khatun, H., Tripathy, M.R. Point and interval estimation of quantiles of several exponential populations with a common location under progressive censoring scheme. Comput Stat 39, 2217–2257 (2024). https://doi.org/10.1007/s00180-023-01410-z

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Received: 08 September 2022
Accepted: 14 August 2023
Published: 14 September 2023
Issue Date: June 2024
DOI: https://doi.org/10.1007/s00180-023-01410-z