The decreasing percentile residual life ageing notion

Earlier researchers have studied some aspects of the classes of distribution functions with decreasing α-percentile residual life (DPRL(α)), 0<α<1. The purpose of this paper is to note some further properties of these classes, and to initiate a theory of non-parametric statistical estimation of DPRL(α) functions. Specifically, the close relationship between the DPRL(α) and the increasing failure rate ageing notions is studied. Other close relationships, between the DPRL(α) ageing notions and the percentile residual life stochastic orders, are described, and further properties of the above classes of distributions are derived. Finally, we introduce an estimator of the percentile residual life function, under the condition that it decreases, and we prove its strongly uniform consistency.