Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of themrespectively. In this paper, we introduce, by taking into account the age of systems, a time-dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ) := P(X t >Y t ) which enables one to obtain, at time t , the probability that the residual lifetime X t is greater than the residual lifetime Y t . It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability , 22, 1973) that the probability of type R(t ) is important for designing as long-lived a product as possible. Several properties of R(t ) and its connection with well-known reliability measures are investigated. The estimation of R(t ) based on samples from X and Y is also discussed.
The cumulative residual entropy (CRE) is a new measure of information and an alternative to the Shannon differential entropy in which the density function is replaced by the survival function. This new measure overcomes deficiencies of the differential entropy while extending the Shannon entropy from the discrete random variable cases to the continuous counterpart. Some properties of the cumulative residual entropy, its estimation and applications has been studied by many researchers. The objective of this paper is twofold. In the first part, we give a central limit theorem result for the empirical cumulative residual entropy based on a right censored random sample from an unknown distribution. In the second part, we use the CRE of the comparison distribution function to propose a goodness-of-fit test for the exponential distribution. The performance of the test statistic is evaluated using a simulation study. Finally, some numerical examples illustrating the theory are also given.
This article considers the properties of a nonparametric estimator developed for a reliability function which is used in many reliability problems. Properties such as asymptotic unbiasedness and consistency are proven for the estimator and using U-statistics, weak convergence of the estimator to a normal distribution is shown. Finally, numerical examples based on an extensive simulation study are presented to illustrate the theory and compare the estimator developed in this article with another based directly on the ratio of two empirical distributions studied in Zardasht and Asadi (2010).
Wei [21] has proposed the relative mean residual life function for comparing two lifetime distributions and studied its properties and relationship with other stochastic orders. In this paper, we obtain some new results on the relative mean residual life function and give a characterization result for a relative ordering based on this function. Motivated by this notion, we also introduce two notions of the dynamic relative cumulative residual entropy functions. Their properties and relationship with other relative orderings are investigated.
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