A two‐parameter parsimonious bathtub model for the analysis of system failure times with illustrations to reliability data

Abstract

AbstractIn this study, a two‐parameter, upper‐bounded probability distribution called the tau distribution is introduced and its applications in reliability engineering are presented. Each of the parameters of the tau distribution has a clear semantic meaning. Namely, one of them determines the upper bound of the distribution, while the value of the other parameter influences the shape of the cumulative distribution function. A remarkable property of this new probability distribution is that its probability density function, survival function, hazard rate function (HRF), and quantile function can all be expressed in terms of its cumulative distribution function. The HRF of the proposed probability distribution can exhibit an increasing trend and various bathtub shapes with or without a low and long‐flat phase (useful time phase), which makes this new distribution suitable for modeling a wide range of real‐world problems. The constraint maximum likelihood estimation, percentile estimation, approximate Bayesian computation, and approximate quantile estimation computation are proposed to calculate the unknown parameters of the model. The suitability of the estimation methods is verified with the aid of simulation and real‐world data results. The modeling capability of the tau distribution was compared with that of some well‐known two‐ and three‐parameter probability distributions using two data sets known from the literature of reliability engineering: time between failures data of a machining center, and time to failure of data acquisition system cards. Based on empirical results, the new distribution may be viewed as a viable competitor to the Weibull, Gamma, Chen, and modified Weibull distributions.