Hazard rate estimation for location-scale distributions under complete and censored data

Abstract

In reliability and life testing studies, the topic of estimating hazard rate has received great attention in recent years since an estimate of hazard rate is a quite useful tool for making decisions. Some works have included nonparametric approaches while some have considered parametric structural models for complete as well as censored data sets; see Meeker et al. (1992), Antoniadis and Gr\'egoire (1999), Rai and Singh (2003), Bezandry et al. (2005), Brunel and Comte (2008), and Mahapatra et al. (2012). Depending on the shapes of the hazard rate, efficiencies differ markedly across proposed estimators. This situation is remarkable especially when different estimation techniques are utilized for unknown parameters of underlying distributions in parametric approaches. That is, estimated hazard rate (and also reliability) at a specific time point t as functions of these estimators leads to inconsistent coverage probabilities as distributional convergence of hazard rate estimator may not be at the desired rate for certain sample sizes. This manuscript focuses on the estimation of monotone hazard rate when the distribution of concern is of location-scale type. A very simple and efficient approximation of hazard rate for a complete sample is introduced as a function of order statistics, which allows a fast convergence to the asymptotic distribution while achieving highly accurate coverage probabilities for confidence intervals (CI). Efficiency of the method is demonstrated on simulation studies by considering a number of location-scale distributions having various hazard shapes. We also consider the case of censored samples by incorporating the predicted future order statistics into the picture.