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Bayesian Estimation of Stress Strength Reliability from Inverse Chen Distribution with Application on Failure Time Data

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Abstract

In this article, we develop Bayesian estimation procedure for estimating the stress strength reliability R = P [X > Y ] when X (strength) and Y (stress) are the inverse Chen random variables. First, we study some statistical properties of the inverse Chen distribution such as quantiles, mode, stochastic ordering, entropy measure, order statistics and stress strength reliability. Then, we estimate the stress strength parameters and R using maximum likelihood and Bayesian estimations. A symmetric (squared error loss) and an asymmetric (entropy loss) loss functions are considered for Bayesian estimation under the assumption of gamma prior. Since, joint posterior distribution of the model parameters and R involve multiple integrations and have complex form. So, we do not get analytical solution without using any numerical techniques. Therefore, we propose to use Lindley’s approximation and Markov chain Monte Carlo techniques for Bayesian computation. A simulation study is carried out for the proposed Bayes estimators of unknown parameters and compared with the maximum likelihood estimator on the basis of mean squared error. Finally, an empirical illustration based on failure time data is presented to demonstrate the applicability of inverse Chen stress strength model.

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Availability of data and material

https://www.jstor.org/stable/1266340, https://doi.org/10.2307/1266340.

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R-code is provided as per need for the reviewers.

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Acknowledgements

The author would like to thanks editor-in-chief and two anonymous referees whose suggestions led to substantial improvement on the manuscript. Author also grateful thanks to Dr Jitendra Kumar, Department of Statistics, Central University of Rajasthan, Ajmer and Dr Vikas Kumar Sharma, Department of Statistics, Banaras Hindu University, Varanasi, India for improving the initial draft of the manuscript.

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Authors and Affiliations

  1. Department of Community Medicine, Jawaharlal Nehru Medical College, Ajmer, Rajasthan, India

    Varun Agiwal

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  1. Varun Agiwal
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Correspondence to Varun Agiwal.

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Appendix

Appendix

$$\begin{aligned}&\rho _{1}=\frac{\partial \log \pi \left( \varTheta \right) }{\partial \lambda _{1}},\rho _{2}=\frac{\partial \log \pi \left( \varTheta \right) }{\partial \lambda _{2}},\rho _{3}=\frac{\partial \log \pi \left( \varTheta \right) }{\partial \beta },\phi _{1}=\frac{\partial \phi \left( \varTheta \right) }{\partial \lambda _{1}},\phi _{2}=\frac{\partial \phi \left( \varTheta \right) }{\partial \lambda _{2}},\\&\phi _{3}=\frac{\partial \phi \left( \varTheta \right) }{\partial \beta },\phi _{11}=\frac{\partial ^{2}\phi (\varTheta )}{\partial \lambda _{1}^{2}}, \phi _{22}=\frac{\partial ^{2}\phi \left( \varTheta \right) }{\partial \lambda _{2}^{2}},\phi _{33}=\frac{\partial ^{2}\phi (\varTheta )}{\partial \beta ^{2}},\phi _{12}=\phi _{21}=\frac{\partial ^{2}\phi \left( \varTheta \right) }{\partial \lambda _{1}\partial \lambda _{2}},\\&\phi _{13}=\phi _{31}=\frac{\partial ^{2}\phi (\varTheta )}{\partial \beta \partial \lambda _{1}},\phi _{23}=\phi _{32}=\frac{\partial ^{2}\phi (\varTheta )}{\partial \beta \partial \lambda _{2}},L_{113}=L_{311}=L_{131}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta \partial \lambda _{1}^{2}},\\&L_{111}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \lambda _{1}^{3}},L_{123}=L_{321}=L_{312}=L_{213}=L_{231}=L_{132}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta \partial \lambda _{1}\partial \lambda _{2}},\\&L_{222}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \lambda _{2}^{3}},L_{112}=L_{211}=L_{121}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \lambda _{1}^{2}\partial \lambda _{2}}, L_{333}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta ^{3}},\\&L_{223}=L_{232}=L_{322}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta \partial \lambda _{2}^{2}},L_{122}=L_{221}=L_{212}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \lambda _{1}\partial \lambda _{2}^{2}},\\&L_{331}=L_{133}=L_{313}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta ^{2}\partial \lambda _{1}},L_{332}=L_{233}=L_{323}=\frac{\partial ^{3}\log L\left( \varTheta \right) }{\partial \beta ^{2}\partial \lambda _{2}},\\&\left( \begin{array}{ccc} \sigma _{11}&{} \sigma _{12}&{} \sigma _{13}\\ \sigma _{21}&{} \sigma _{22}&{} \sigma _{23}\\ \sigma _{31}&{} \sigma _{32}&{} \sigma _{33}\\ \end{array}\right) =-\left( \begin{array}{ccc} \frac{\partial ^{2}\log L}{\partial \lambda ^{2}_{1}}&{}\frac{\partial ^{2}\log L}{\partial \lambda _{1}\partial \lambda _{2}}&{}\frac{\partial ^{2}\log L}{\partial \lambda _{1}\partial \beta }\\ \frac{\partial ^{2}\log L}{\partial \lambda _{1}\partial \lambda _{2}}&{} \frac{\partial ^{2}\log L}{\partial \lambda ^{2}_{2}}&{} \frac{\partial ^{2}\log L}{\partial \lambda _{2}\partial \beta }\\ \frac{\partial ^{2}\log L}{\partial \beta \partial \lambda _{1}}&{} \frac{\partial ^{2}\log L}{\partial \beta \partial \lambda _{2}}&{}\frac{\partial ^{2}\log L}{\partial \beta ^{2}} \end{array}\right) ^{-1}=-\left( \begin{array}{ccc} \delta _{11}&{}\delta _{12}&{}\delta _{13}\\ \delta _{21}&{} \delta _{33}&{} \delta _{23}\\ \delta _{31}&{}\delta _{32}&{}\delta _{33} \end{array}\right) ^{-1},\\&\rho _{1}=\frac{(a_{1}-1)}{\lambda _{1}}-b_{1},\rho _{2}=\frac{(a_{2}-1)}{\lambda _{2}}-b_{2},\rho _{3}=-\frac{1}{\beta },\delta _{11}=-\frac{n}{\lambda _{1}^{2}}, \delta _{12}=\delta _{21}=0, \\&\delta _{22}=-\frac{m}{\lambda _{2}^{2}},\delta _{13}=\delta _{31}=\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})e^{x_{i}^{-\beta }},\delta _{23}=\delta _{32}=\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})e^{y_{j}^{-\beta }},\\&\delta _{33}=-\frac{\left( n+m \right) }{\beta ^{2}}+\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})^{2}-\lambda _{1}\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})^{2}e^{x_{i}^{-\beta }}\left( 1+x_{i}^{-\beta }\right) +\\&\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})^{2}-\lambda _{2}\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})^{2}e^{y_{j}^{-\beta }}\left( 1+y_{j}^{-\beta }\right) , L_{111}=\frac{2n}{\lambda _{1}^{3}}, L_{222}=\frac{2m}{\lambda _{2}^{3}},\\&L_{333}=\frac{2\left( n+m \right) }{\beta ^{3}}-\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})^{3}-\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})^{3}+\lambda _{1}\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})^{3}\\&e^{x_{i}^{-\beta }}\left( 1+3x_{i}^{-\beta }+x_{i}^{-2\beta }\right) +\lambda _{2}\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})^{3}e^{y_{j}^{-\beta }}\left( 1+3y_{j}^{-\beta }+y_{j}^{-2\beta }\right) , \\&L_{133}=L_{331}=L_{313}=-\sum \limits _{i=1}^{n}x_{i}^{-\beta }(\log x_{i})^{2}e^{x_{i}^{-\beta }}\left( 1+x_{i}^{-\beta }\right) ,L_{233}=L_{332}=L_{323}\\&=-\sum \limits _{j=1}^{m}y_{j}^{-\beta }(\log y_{j})^{2}e^{y_{j}^{-\beta }}\left( 1+y_{j}^{-\beta }\right) ,L_{123}=L_{321}=L_{312}=L_{213}=L_{231}=\\&L_{132}=L_{113}=L_{311}=L_{131}=L_{112}=L_{211}=L_{121}=L_{223}=L_{232}=L_{322}=\\&L_{122}=L_{221}=L_{212}=0,R=\frac{\lambda _{1}}{\lambda _{1}+\lambda _{2}},R_{1}=\frac{\lambda _{2}}{\left( \lambda _{1}+\lambda _{2} \right) ^{2} }, R_{2}=-\frac{\lambda _{1}}{\left( \lambda _{1}+\lambda _{2} \right) ^{2} },\\&R_{11}=-\frac{2\lambda _{2}}{\left( \lambda _{1}+\lambda _{2} \right) ^{3} }, R_{22}=\frac{2\lambda _{1}}{\left( \lambda _{1}+\lambda _{2} \right) ^{3} },R_{12}=\frac{\lambda _{1}-\lambda _{2}}{\left( \lambda _{1}+\lambda _{2} \right) ^{3} },R^{-1}=1+\frac{\lambda _{2}}{\lambda _{1}},\\&R_{1}^{-1}=-\frac{\lambda _{2}}{\lambda _{1}^{2}}, R_{2}^{-1}=\frac{1}{\lambda _{1}}, R_{11}^{-1}=\frac{2\lambda _{2}}{\lambda _{1}^{3}}, R_{22}^{-1}=0, R_{12}^{-1}=-\frac{1}{\lambda _{1}^{2}}. \end{aligned}$$

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Agiwal, V. Bayesian Estimation of Stress Strength Reliability from Inverse Chen Distribution with Application on Failure Time Data. Ann. Data. Sci. 10, 317–347 (2023). https://doi.org/10.1007/s40745-020-00313-w

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