Bayes Risk of Mean Residual with System Reliability and Empirical Application

The Bayes risk of the mean residual (MR) or excess of a random variable above a threshold is defined by the expectation with respect to a prior for the threshold. Recently the MR Bayes risk has been studied for the proportional hazards (PH) model. We provide results without assuming any stricture such as PH for the distribution, further results for the PH case and new reliability applications, and an empirical application. We show that the standard deviation provides a tight upper bound for the MR Bayes risk. Attainment of the bound characterizes the exponential distribution. The ratio of the MR Bayes risk to the standard deviation provides a risk index which reaches the maximum of one if and only if the underlying distribution is exponential. Distributional examples illustrate the tightness of the standard deviation bound and the improvement over the known bound. Another result broadens the notion of MR Bayes risk to functions of the threshold. We provide a result for ordering of PH models by MR Bayes risk. New reliability applications include the MR Bayes risk at the system level and distributions with new better (worse) than used in expectation property. An empirical example illustrates application of the MR and MR Bayes risk for the New York City’s taxi trip times.