Estimation and prediction using classical and Bayesian approaches for Burr III model under progressive type-I hybrid censoring

Abstract

In this paper we address the problems of estimation and prediction when lifetime data following Burr type III distribution are observed under progressive type-I hybrid censoring. We first obtain maximum likelihood estimators of unknown parameters using expectation maximization and stochastic expectation maximization algorithms, and associated interval estimates using Fisher information matrix. We then obtain Bayes estimators based on non-informative and informative priors under squared error, entropy and Linex loss functions using the method of Tierney–Kadane and importance sampling technique, and associated highest posterior density interval estimates by making use of Chen and Shao method. We further predict the censored observations and interval estimates under classical and Bayesian approaches. Finally we analyze two real data sets, and conduct a simulation study to compare the performance of various proposed estimators and predictors.

Appendix

$$\begin{aligned}&E_{1i}(\alpha , \beta ) \\&\quad = E\big (\ln (1+z_{(ij)}^{-\beta }) \mid z_{(ij)}> x_{(i)}\big ) \\&\quad = \frac{\alpha }{1-F(x_{(i)}; \alpha , \beta )} \int _{1}^{1+x_{(i)}^{-\beta }} u^{-(\alpha +1)} \ln u \, du,\\&E_{1T}(\alpha , \beta ) \\&\quad = E\big (\ln (1+z_{(Tj)}^{-\beta }) \mid z_{(Tj)}> T\big ) \\&\quad = \frac{\alpha }{1-F(T; \alpha , \beta )} \int _{1}^{1+T^{-\beta }} u^{-(\alpha +1)} \ln u \, du,\\&E_{2i}(\alpha , \beta ) \\&\quad = E\big (\ln z_{(ij)} \mid z_{(ij)}> x_{(i)}\big ) \\&\quad =\frac{\alpha }{\beta (1-F(x_{(i)}; \alpha , \beta ))}\\&\qquad \int _{1}^{1+x_{(i)}^{-\beta }} u^{-(\alpha +1)} \ln (u-1) \, du,\\&E_{2T}(\alpha , \beta ) \\&\quad = E\big (\ln z_{(Tj)} \mid z_{(Tj)}> T\big )\\&\quad = \frac{\alpha }{\beta (1-F(T; \alpha , \beta ))}\\&\qquad \int _{1}^{1+T^{-\beta }} u^{-(\alpha +1)} \ln (u-1) \, du,\\&E_{3i}(\alpha , \beta ) \\&\quad = E\Big (\frac{z_{(ij)}^{-\beta } \ln z_{(ij)}}{1+z_{(ij)}^{-\beta } } \mid z_{(ij)}> x_{(i)}\Big ) \\&\quad = \frac{\alpha }{\beta (1-F(x_{(i)}; \alpha , \beta ))}\\&\quad \int _{1+x_{(i)}^{-\beta }}^{1} u^{-(\alpha +2)} (u-1)\ln (u-1) \, du,\\&E_{3T}(\alpha , \beta ) \\&\quad = E\Big (\frac{z_{(Tj)}^{-\beta } \ln z_{(Tj)}}{1+z_{(Tj)}^{-\beta } } \mid z_{(Tj)} > T\Big ) \\&\quad =\frac{\alpha }{\beta (1-F(T; \alpha , \beta ))} \\&\qquad \int _{1+T^{-\beta }}^{1} u^{-(\alpha +2)} (u-1)\ln (u-1) \, du,\\&e_{1}(\mathcal {C}^*; \alpha , \beta ) \\&\quad = \frac{\alpha }{\beta (1-F(\mathcal {C}^* ; \alpha , \beta ))}\\&\qquad \int _{1}^{1+{\mathcal {C}^*}^{-\beta }}(u-1)\ln (u-1) u^{-(\alpha +2)} du,\\&e_{2}(\mathcal {C}^*; \alpha , \beta ) \\&\quad = \frac{\alpha {\mathcal {C}^*}^{-\beta }(1+{\mathcal {C}^*}^{-\beta })^{-(\alpha +1)}\ln \mathcal {C}^*}{\beta [1-F(\mathcal {C}^* ; \alpha , \beta )]^2}\\&\qquad \Big ( 1-F(\mathcal {C}^* ; \alpha , \beta ) - \alpha \ln (1+{\mathcal {C}^*}^{-\beta })\Big ),\\&e_{3}(\mathcal {C}^*; \alpha , \beta ) \\&\quad = \frac{\alpha }{\beta ^{2}[1-F(\mathcal {C}^* ; \alpha , \beta )]}\\&\qquad \int _{1}^{1+{\mathcal {C}^*}^{-\beta }} (u-1)(\ln (u-1))^{2} u^{-(\alpha +3)}du,\\&e_{4}(\mathcal {C}^*; \alpha , \beta ) \\&\quad = \frac{\alpha {\mathcal {C}^*}^{-\beta }(1+{\mathcal {C}^*}^{-\beta })^{-(\alpha +2)}\ln \mathcal {C}^*}{\beta [1-F(\mathcal {C}^* ; \alpha , \beta )]^{2}}\\&\qquad \Big (F(\mathcal {C}^* ; \alpha , \beta )-1+\alpha {\mathcal {C}^*}^{-\beta }\Big ). \end{aligned}$$