On empirical cumulative residual entropy and a goodness-of-fit test for exponentiality

Abstract

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.