Log-Odds Rate and Monotone Log-Odds Rate Distributions

Abstract

Since the 1960's, reliability models for time to failure based on monotone failure rate models have become important models of failure time for reliability practitioners. Bounds for monotone increasing failure rates (IFR) have been developed and are especially useful for bounding the hazard of aging. Recently, however, it appears that the IFR model as a model of aging may be too stringent for some modern components. This supposition is confirmed by the widespread use of the log normal model to describe time to failure. This paper introduces a new time-to-failure model based on the log-odds rate (LOR) which is comparable to the model based on the failure rate. It is shown that failure-time distributions can be characterized by LOR and that the increasing LOR (ILOR) model in terms of log time (ℓn(t)) is less stringent than the IFR model for aging. It is shown that the logistic distribution has the property of a constant LOR and that the log-logistic distribution has the property of a constant LOR with respect to ℓn(t). Some properties of ILOR distributions are presented and bounds based on the relationship to the log-logistic distribution are provided for distributions which are ILOR with respect to ℓn(t). Examples of the use of the bounds are also presented, and examples of the comparisons of the ILOR bounds with the IFR bounds are made.