Modeling Discrete Bathtub and Upside Down Bathtub Mean Residual Life Functions.

A useful function for analyzing burn-in, developing maintenance policies, or simply modeling lifetime of equipment is the mean residual life function. Other functions, such as the reliability or the failure rate functions, are of course important also. Discrete data arises naturally in various ways from discretizing or grouping continuous data, devices operate by cycles e.g., a copiers cycle is a copy, its lifelength the total number of copies, etc. This paper develops a general approach to modeling discrete bathtub and upside down bathtub mean residual life functions. Because the approach allows parametric modeling of the mean residual life maximum likelihood estimation of models can be done. This will enable estimation of such parametric models for complete discrete data, as well as right censored discrete data. A simple, perhaps surprising, example is presented where the mean residual life increases, then decreases however, the hazard rate also increases, drops suddenly at one cycle, then increases again. The authors discuss two reasonable industrial explanations of such unusual behavior.