Bayesian inference of a dependent competing risk data

Abstract

Recently, Feizjavadian and Hashemi (Analysis of dependent competing risks in presence of progressive hybrid censoring using Marshall–Olkin bivariate Weibull distribution. Comput Stat Data Anal. 2015;82:19–34) provided a classical inference of a competing risks data set using Marshall–Olkin bivariate Weibull distribution when the failure of an unit at a particular time point can happen due to more than one cause. The aim of this paper is to provide the Bayesian analysis of the same model based on a very flexible Gamma–Dirichlet (GD) prior on the scale parameters. The Bayesian inference has certain advantages over the classical inference in this case. We provide the Bayes estimates of the unknown parameters and the associated highest posterior density credible intervals based on Gibbs sampling technique. We further consider the Bayesian inference of the model parameters assuming partially ordered GD prior on the scale parameters when one cause is more severe than the other cause. We have extended the results for different censoring schemes also.

Keywords:

• Marshall–Olkin bivariate Weibull distribution
• Gamma–Dirichlet distribution
• Bayes estimates
• competing risk
• order restricted inference

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix

Proof of Theorem 1

$\begin{array}{rl}\ln (\widetilde{{\pi }_1}(\alpha ))& =-c_1\alpha +(n+c_2-1)\ln (\alpha )-(a+n)\ln (b+\sum _{i=1}^n{t}_{i:n}^{\alpha })\\ & +(\alpha -1)\sum _{i=1}^n\ln (t_{i:n}),\\ \frac{{\mathrm{\partial }}^2\ln (\widetilde{{\pi }_1}(\alpha ))}{\mathrm{\partial }{\alpha }^2}& =-\frac{n+c_2-1}{{\alpha }^2}-(a+n)\\ & \times \left[\frac{\begin{array}{l}b\sum _{i=1}^n{t}_{i:n}^{\alpha }(\ln (t_{i:n}){)}^2+\sum _{i=1}^n{t}_{i:n}^{\alpha }\sum _{i=1}^n{t}_{i:n}^{\alpha }(\ln (t_{i:n}){)}^2\\ -(\sum _{i=1}^n{t}_{i:n}^{\alpha }\ln (t_{i:n}){)}^2\end{array}}{(b+\sum _{i=1}^n{t}_{i:n}^{\alpha }{)}^2}\right]\\ & \le 0,\end{array}$since $\sum _{i=1}^n{t}_{i:n}^{\alpha }\sum _{i=1}^n{t}_{i:n}^{\alpha }(\ln (t_{i:n}){)}^2-(\sum _{i=1}^n{t}_{i:n}^{\alpha }\ln (t_{i:n}){)}^2\ge 0$ (by Cauchy–Schwarz inequality).