Remarks on characterizations of Malinowska and Szynal

Abstract

The problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution.To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. In this work, several characterizations of Malinowska and Szynal (2008) for certain general classes of distributions are revisited and simpler proofs of them are presented. These characterizations are not based on conditional expectation of the kth lower record values (as in Malinowska and Szynal), they are based on: (i) simple truncated moments of the random variable, (ii) hazard function.

Some Characterization and Relations Based on K-th Lower Record Values

“…Moreover, Theorems 2.5 and 2.6 in Shawky and Abu-Zinadah [18] and Theorems 3 and 4 in Shawky and Bakoban [19] are generalized as shown in Theorems 2 and 3 using the cdf in (7) by characterizing the second general class of distributions through conditional expectation of k-th lower record values. Lastly, we show that equation (2.1.1) in Hamedani, Javanshiri, Maadooliat, and Yazdani [20] is a special case of Theorem 4 by using the cdf in (8) as the third general class of distributions based on truncated moments of some random variable. Some distributions as members of these general classes are given as examples in Tables 1 and 2.…”

“…The RHS of equation (20) where a = is the same as that of equation (2.1.1) in Hamedani, Javanshiri, Maadooliat, and Yazdani [20].…”

The Kumaraswamy Marshal-Olkin family of distributions

“…The following proposition has already appeared in [31] Theorem 2.1.3, so we will just state it here for the sake of completeness.…”

“…The log-likelihood function can be maximized either directly by using the R (AdequecyModel), SAS (PROC NLMIXED) or the Ox program (sub-routine MaxBFGS) [32] or by solving the nonlinear likelihood equations obtained by differentiating (31). The score function U n ðHÞ ¼ @' n =@a; @' n =@b; @' n =@p; ð @' n =@nÞ > has components given by where h ðnÞ ðÁÞ means the derivative of the function h with respect to n. The observed information matrix can be obtained from the authors under request.…”

Characterizations of Distributions Via Conditional Expectation of Function of Generalized Order Statistics

“…Hamedani et al [4] presented the following characterization result, among others, in the spirit of the above two characterizations. …”