Characterizations Based on Cumulative Entropy of the Last-order Statistics

Cumulative Entropy (CE) as a measure of uncertainty alternative to Shannon entropy is proposed by Di Crescenzo and Longobardi (2009). In this paper, some properties of the cumulative entropy are derived and under conditions are showed the cumulative entropy of the last order statistics can determine the distribution function uniquely. Weibull family is characterized by ratio of the cumulative entropy of the last order statistics to its expectation. Also, some inequalities are presented for the cumulative entropy of reversed residual lifetime of a parallel system.


Introduction
introduced entropy, known Shannon entropy, as a measure of uncertainty for a random variable.Suppose  be a non-negative continuous random variable having probability density function  and cumulative distribution function .Hence, Shannon entropy ℎ() of a continuous non-negative random variable  with probability density function () is defined as It's clear that H(X; ∞) = ℎ().Rao et al. (2004) defined Cumulative Residual Entropy(CRE) as an alternative measure of uncertainty to Shannon entropy in that the probability density function is replaced by survival(reliability) function and obtained some properties and applications of that(2005).Rao showed CRE overcomes some problems with Shannon entropy.Some properties of CRE are: CRE is more general rather than the Shannon entropy, it has more mathematical properties rather than Shannon entropy, and it is easily computed from sample data.CRE has applications in reliability engineering and image processing, for more details see Rao (2004Rao ( , 2005)).CRE for a non-negative univariate random variable is as follows: Crescenzo and Longobardi (2009) introduced a new information measure that turns out to be useful to measure information on the reversed residual lifetime of a system.The reversed residual lifetime is a concept in reliability that is convenient to describe the time elapsing between the failure of a system and the time when it is down.This measure is defined as follows: where () is the cumulative distribution function of the non-negative random variable .Also, they proposed dynamic form of cumulative entropy that called dynamic cumulative entropy and obtained some of its properties.By using probability integral transformation () =  that  have uniform distribution function on (0,1), then we obtain Suppose  1 ,  2 , … ,   be a random sample, the order statistics of the sample is defined by the arrangement  1 ,  2 , … ,   from the minimum to the maximum by  1: ,  2: , … ,  : .Order statistics are used in entropy estimation, quality control, reliability, insurance, etc.For more details see Arnold et al. (1992), David and Nagaraja (2003).
Baratpour ( 2010) have derived characterizations result based on Cumulative residual entropy of first order statistics.Also Thapliyal et al. (2013) studied cumulative entropy and dynamic cumulative entropy using order statistics.They showed a characterization property that dynamic cumulative entropy of the i th order statistics determines the distribution function.
The purpose of this paper is characterization result the parent distribution based on Cumulative Entropy of the last order statistics.
The paper is organized as follows: In section 2, characterizing the parent distributions based on the CE of the last order statistics is presented.Also, Weibull distribution is characterized based on the ratio of the CE of the last order statistics to the expectation of it.Last section includes some Characterizations based on CE of reversed residual lifetime of parallel systems.

Characterizations Based on the Last-order Statistics
Last order statistics is an important special case of order statistics.Suppose  : be the last order statistics in a random sample of size n, cumulative distribution function of  : is   : () = (())  , Thus CE of  : is defined as follow: By changing variable () = , we have In the following theorem, we show that only in Weibull family the ratio of the CE of the last order statistics to the expectation of it is constant.First we need a lemma taken from Baratpour (2010).Relation (4) coincides c, so we have = ∞, then from lemma 2.1 we have then it follows: By solving this differential equation, we have ) 1 c } , x > 0 we conclude that F(x) belongs to the Weibull family.
Theorem 2.2: Suppose ,  be two positive random variables with probability density functions () and () and absolutely continuous distribution functions () and (), respectively.Then  and  belong to the same family of distributions, but for a location shift, if and only if be infinitely.
Proof: The 'if' part of the theorem is trivial.Hence we prove 'only if' part, according to relation (2) if ( : ) = ( : ), then = ∞, then from Lemma 2.1 we can conclude that means for a change of location parameter,  and  belong to the same family.

Characterizations Based on Cumulative Entropy of Reversed Residual Lifetime of Parallel Systems
Suppose  1 ,  2 , … ,   be continuous and iid random variables with common cumulative distribution function  denote the lifetimes of  components of a parallel system.Also  1: ,  2: , … ,  : be the ordered lifetimes of the components.Then  : represents the lifetime of that parallel system with cumulative distribution function Longbardi (2002)  introduced the entropy of the reversed residual lifetime   = [ − | < ] as a dynamic measure of uncertainty as follows (; []) Cumulative entropy for reversed residual lifetime with distribution function   () = [ − | < ] is (; ) = − ∫   ()  ()  = clear (; ∞) = ().