Characterizations of Distributions Based on Properties of Exponentially Shifted Records

Abstract

Let ξ1, ξ2, ... be independent exponentially E(1)-distributed r.v.’s with the probability density function r(x)=exp(-x), x ≥0, and the distribution function R(x)=max{0,1-exp(-x)}. The special case of record values based on r.v.’s ξ1, ξ2, ... will be denoted as Z(1)<Z(2)<...<Z(n)<... . Record results in various spheres of human activities are very popular among citizens all over the world. It is enough to remember a lot of editions of the Guinness Book of Records. Therefore it is not surprising that there is the great interest among the specialists in probability and mathematical statistics to study and to solve the various problems which arise in the mathematical theory of records. There are a lot of monographs and papers (see, for example, references [1]-[9]), where the different aspects of the theory of records are considered. Meanwhile up to now there exist some of directions of this theory which attract the attention of mathematicians. One of these interesting directions is connected with obtaining the new characterizations of probability distributions based on the different properties of record values. Some of the corresponding results are given below.