Abstract

Although there has been an increasing interest in one-parameter lifetime distributions used in reliability modeling, there has been a lack of investigation on statistical inference of the Pranav distribution under a progressive first-failure censoring scheme. In this paper, we aim to address this issue by proposing classical and Bayesian estimation methods for the unknown parameter theta, the reliability function R(x), and the hazard rate function h(x) of the Pranav distribution based on a progressively first-failure censoring scheme. The maximum likelihood estimation of the Pranav distribution parameters is implemented by the BFGS algorithm, and the asymptotic confidence intervals of the parameters are obtained by the normal approximation and log transformation methods. The delta method is used to compute the standard errors of the reliability function and the hazard rate function of the Pranav distribution. On the other hand, the random walk Metropolis-Hastings algorithm is implemented within a Bayesian estimation framework of the Pranav distribution parameters under two different gamma priors, a non-informative prior Gamma(0.001, 0.001) and an informative prior Gamma(2, 2). The Bayesian credible intervals and the highest posterior density intervals of the Pranav distribution parameters are calculated by the posterior samples obtained from the random walk Metropolis-Hastings algorithm, and the parametric bootstrap confidence intervals are given as an alternative solution.

The Monte Carlo simulation study used nine censoring plans, which vary according to sample size (n = 20, 30, 50), censoring proportion (m = 27, 22, 18 out of n = 30), and group size (k = 3, 5, 7), and evaluates the performance of the estimators for root mean squared error (RMSE), mean absolute bias (MAB), average interval length (AIL), and coverage probability (CP). The RMSE of theta_MLE decreases from 0.6206 for n = 20 to 0.5549 for n = 50 under Plan I. The decrease is 10.6%. The informative prior always results in the smallest RMSE for all nine plans. The range varies from 0.4536 to 0.5976. The coverage probability for the ACI-NA interval for theta varies from 0.267 to 0.733. The HPD interval performs comparably. Two real-life examples are used to illustrate the application. The first example deals with failures of Kevlar 49/epoxy strands, and the second deals with air conditioners. In the case of air conditioners, the MLE and Bayesian posterior mean using Prior 1 agree to within 0.58%. In contrast, Prior 2 gives a very different result. The Kolmogorov-Smirnov test gives p = 0.0035 for the Kevlar example and p = 0.0001 for the air conditioners. Both examples, however, appear to have a reasonable fit. Model selection criteria used are AIC, BIC, and DIC.