G and related distributions with applications in reliability growth analysis

摘要

Motivated by four unsolved issues on the mean time between failures (MTBFs) in nonhomogeneous Poisson processes (NHPP) with power law intensity function for complete/ incomplete observations, in this article, we first study some important properties on three new distributions (i.e., the G, inverse G, and RG distributions). Next, we develop three methods (i.e., the Lagrange multiplier, quantile-based and sampling-based methods) to establish the shortest confidence intervals for the MTBF in a single repairable system and for the MTBF ratio in two independent repairable systems; and also develop two methods (i.e., the density-based and sampling-based methods) within the framework of the critical region and $p$-value approaches to test hypotheses on the MTBF and the MTBF ratio. Simulation studies are performed to compare the proposed methods. Two real data sets are used to illustrate the proposed statistical methods.