Estimation of Lifetime Performance Index for Generalized Inverse Lindley Distribution Under Adaptive Progressive Type-II Censored Lifetime Test

The lifetime performance index (LPI) is an important metric for evaluating product quality, and research on the statistical inference of the LPI is of great significance. This paper discusses both the classical and Bayesian estimations of the LPI under an adaptive progressive type-II censored lifetime test, assuming that the product's lifetime follows a generalized inverse Lindley distribution. At first, the maximum likelihood estimator of the LPI is derived, and the Newton-Raphson iterative method is adopted to solve the numerical solution due to the log-likelihood equations having no analytical solutions. If the exact distribution of the LPI is not available, then the asymptotic confidence interval and bootstrap confidence interval of the LPI are constructed. For the Bayesian estimation, the Bayesian estimators of the LPI are derived under three different loss functions. Due to the complex multiple integrals involved in these estimators, the MCMC method is used to draw samples and further construct the HPD credible interval of the LPI. Finally, Monte Carlo simulations are used to observe the performance of these estimators in terms of the average bias and mean squared error, and two practical examples are used to illustrate the application of the proposed estimation method.