Classical and Bayesian estimation of the reliability function for the inverse Lindley distribution based on lower record statistics

The reliability function‎, ‎or the survival function at a specified time t‎, ‎denotes the proportion of products that remain operational beyond time t and continue to function‎. ‎This interpretation underscores the pivotal role of the survival function and its estimation in understanding lifetime phenomena‎. ‎This paper explores the estimation of the survival function for the inverse Lindley distribution based on lower records‎. ‎The estimation techniques encompass maximum likelihood and bootstrap methods‎. ‎Furthermore‎, ‎Bayesian approaches employing Metropolis-Hastings and importance sampling algorithms are employed‎. ‎In addition to deriving approximate confidence intervals using the delta method and percentile bootstrap intervals for the survival function‎, ‎Chen and Shao's shortest width credible intervals are also determined‎. ‎A comprehensive simulation study is presented to assess the effectiveness of both point and interval estimators‎. Finally‎, ‎an application of the results is given to a real data set.