Gamma lifetimes and one-shot device testing analysis

Abstract

Gamma distribution is widely used to model lifetime data in reliability and survival analysis. In the context of one-shot device testing, encountered commonly in testing devices such as munitions, rockets, and automobile air-bags, either left- or right-censored data are collected instead of actual lifetimes of the devices under test. The destructive nature of one-shot devices makes it difficult to collect sufficient lifetime information on the devices. For this reason, accelerated life-tests are commonly used in which the test devices are subjected to conditions in excess of its normal use-condition in order to induce more failures, so as to obtain more lifetime information within a relatively short period of time. In this paper, we discuss the analysis of one-shot device testing data under accelerated life-tests based on gamma distribution. Both scale and shape parameters of the gamma distribution are related to stress factors through log–linear link functions. Since lifetimes of devices under this test are censored, the EM algorithm is developed here for the estimation of the model parameters. The inference on the reliability at a specific mission time as well as on the mean lifetime of the devices is also developed. Moreover, by using missing information principle, the asymptotic variance–covariance matrix of the maximum likelihood estimates under the EM framework is determined, and is then used to construct asymptotic confidence intervals for the parameters of interest. For the reliability at a specific mission time and also for the mean lifetime of the devices, transformation approaches are proposed for the construction of confidence intervals. These confidence intervals are then compared through a simulation study in terms of coverage probabilities and average widths. Recommendations are then made for an appropriate approach for the construction of confidence intervals for different sample sizes and different levels of reliability. A distance-based statistic is suggested for testing the validity of the model to an observed data. Finally, since current status data with covariates in survival analysis and one-shot device testing data with stress factors in reliability analysis share the same data structure, a real data from a toxicological study is used to illustrate the developed methods.