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 nonparametric statistical estimation of decreasing α-percentile residual life functions. Specifically, the close relationship between the DPRL(α) and the IFR (increasing failure rate) aging notions is studied. Other close relationships, between the DPRL(α) aging 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.
The overlap coefficient ([Formula: see text]) measures the similarity between two distributions through the overlapping area of their distribution functions. Given its intuitive description and ease of visual representation by the straightforward depiction of the amount of overlap between the two corresponding histograms based on samples of measurements from each one of the two distributions, the development of accurate methods for confidence interval construction can be useful for applied researchers. The overlap coefficient has received scant attention in the literature since it lacks readily available software for its implementation, while inferential procedures that can cover the whole range of distributional scenarios for the two underlying distributions are missing. Such methods, both parametric and non-parametric are developed in this article, while R-code is provided for their implementation. Parametric approaches based on the binormal model show better performance and are appropriate for use in a wide range of distributional scenarios. Methods are assessed through a large simulation study and are illustrated using a dataset from a study on human immunodeficiency virus-related cognitive function assessment.
In this paper we study a family of stochastic orders of random variables defined via the comparison of their percentile residual life functions. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory and finance are described.
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