Certain Characterizations of Recently Introduced Distributions

Various characterizations of twenty four recently introduced distributions are presented. These characterizations are based on: ( ) ratio of two truncated moments; ( ) the hazard function and ( ) conditional expectations.


Introduction
In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution.To this end, the investigator will rely on the characterizations of the selected distribution.Generally speaking, the problem of characterizing a distribution is an important problem in various fields and has recently attracted the attention of many researchers.Consequently, various characterization results have been reported in the literature.These characterizations have been established in many different directions.For detailed treatments and the domain of applicability of each of these distributions, we refer the interested reader to the corresponding papers cited in the References section.
We list below the cumulative distribution function (cdf) and probability density function (pdf) of each one of these distributions in the same order as listed above.We will be employing the same notation for the parameters as chosen by the original authors.
A) The cdf and pdf of (H-GC) are given, respectively, by K) The cdf and pdf of (TG) are given, respectively, by L) The cdf and pdf of (EBXII) are given, respectively, by where are parameters.
M) The cdf and pdf of (WFr) are given, respectively, by where are all positive parameters.
N) The cdf and pdf of (MOP W ) are given, respectively, by O) The cdf and pdf of (TEWG) are given, respectively, by where , ) , | | are parameters.
P) The cdf and pdf of (TGG) are given, respectively, by R) The cdf and pdf of (MOEGR) are given, respectively, by where are parameters and ( ) ( ) ∫ S) The cdf and pdf of (GIL) are given, respectively, by where are parameters.
T) The cdf and pdf of (Kw-TEAW) are given, respectively, by U) The cdf and pdf of (BEG ) are given, respectively, by where and are all positive parameters.
V) The cdf and pdf of (KwKwW) are given, respectively, by X) The cdf and pdf of (EPL) are give, respectively, by where , both positive and are parameters.

Characterization Results
As mentioned in the Introduction, characterizations of distributions is an important research area which has recently attracted the attention of many researchers.This section deals with various characterizations of the distributions listed in the Introduction.These characterizations are based on: ( ) a simple relationship between two truncated moments; ( ) the hazard function; ( ) conditional expectation of a single function of the random variable.It should be mentioned that for the characterization ( ) the cdf need not have a closed form and depends on the solution of a first order differential equation, which provides a bridge between probability and differential equation.

Characterizations based on two truncated moments
In this subsection, we present characterizations of all the distributions mentioned in the Introduction, in terms of a simple relationship between two truncated moments.Our first characterization result employs a theorem due to (Glänzel, 1987), see Theorem 2.1.1 below.Note that the result holds also when the interval is not closed.Moreover, as mentioned above, it could be also applied when the cdf does not have a closed form.As shown in (Glänzel, 1990), this characterization is stable in the sense of weak convergence.
Theorem 2.1.1.Let ( ) be a given probability space and let , be an interval for some ( ) Let be a continuous random variable with the distribution function and let and be two real functions defined on such that is defined with some real function .Assume that ( ), ( ) and is twice continuously differentiable and strictly monotone function on the set .Finally, assume that the equation has no real solution in the interior of .Then is uniquely determined by the functions and , particularly where the function is a solution of the differential equation and is the normalization constant, such that ∫ .
Here is our first characterization.
The general solution of the differential equation ( ) is where is a constant.Note that a set of functions satisfying the differential equation ( ) is given in Proposition 2.1.1 with However, it should also be noted that there are other triplets ( ) satisfying the conditions of Theorem 2.1.1.
The proofs of the following Propositions (in this subsection) are similar to that of Proposition 2.1.1,so we only state the Propositions with their corresponding Corollaries.Further, we will not repeat the last sentence of the above paragraph for other distributions.
The general solution of the differential equation ( ) is where is a constant.Note that a set of functions satisfying the differential equation ( ) The general solution of the differential equation ( where is a constant.Note that a set of functions satisfying the differential equation ( ) The general solution of the differential equation ( ) is where is a constant.Note that a set of functions satisfying the differential equation ( ) The general solution of the differential equation ( ) is where is a constant.Note that a set of functions satisfying the differential equation ( ) distribution.( ) Again, clearly, there are other suitable functions as well, we chose the above ones for the sake of simplicity.

Concluding Remarks
In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution.To this end, the investigator will rely on the characterizations of the selected distribution.Consequently, various characterization results have been reported in the literature.These characterizations have been established in many different directions.The present work deals with the characterizations of new univariate continuous distributions which have appeared in the literature in 2015-2016.We certainly hope that the content of this work will be useful to the investigators who are interested to know if they have chosen the right distributions.
For TGG reduces to TG (Transmuted Gompertz) Distribution, discussed by the same authors, which appeared in Pakistan Journal of Statistics, Vol.32, No.3 , 2016.Q) The cdf and pdf of (NBBS) are given, respectively, by given in Proposition 2.1.2with given in Proposition 2.1.3with given in Proposition 2.1.4with given in Proposition 2.1.5with Proposition 2.3.2 provides a characterization of (WFr)