Characterizations of Exponentiated Generalized Power Lindley and Geometric-Zero Truncated Poisson Distributions

MirMostafaee et al. (2019) proposed a continuous univariate distribution called Exponentiated Generalized Power Lindley (EGPL) distribution and studied certain properties and applications of their distribution. Akdogan et al. (2019) introduced a discrete distribution called Geometric-Zero Truncated Poisson (GZTP) distribution and provided its properties and applications. The present short note is intended to complete, in some way, the works cited above via establishing certain characterizations of the EGPL and GZTP distributions in different directions.


Introduction
Characterizations of distributions is an important research area which has recently attracted the attention of many researchers. This short note deals with various characterizations of EGPL and GZTP distributions to complete, in some way, the works of MirMostafaee et al. (2019) and Akdogan et al. (2019). These characterizations are based on: (i) a simple relationship between two truncated moments; (ii) the hazard function; (iii) the reverse hazard function and (iv) the conditional expectation of a function of the random variable. It should be mentioned that for characterization (i) the cd f (cumulative distribution function) is not required to have a closed form. MirMostafaee et al. (2019) proposed the EGPL distribution with cd f and pd f (probability density function) given, respectively, by and where β, λ, a, b are all positive parameters. Akdigan et al. (2019) introduced the GZTP distribution with cd f and pm f (probability mass function) given, respectively, by and where θ ∈ (0, 1) and q ∈ (0, 1) are parameters.
Remark 1.1. Clearly, one can write down the formulas for the hazard and reverse hazard functions corresponding to these distributions as needed.

Characterizations of EGPL Distribution
We present our characterizations of EGPL in four subsections.

Characterizations Based on Two Truncated Moments
In this subsection we present characterizations of EGPL distribution in terms of a simple relationship between two truncated moments. The 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 H is not closed. Moreover, as mentioned above, it could be also applied when the cd f F does not have a closed form. As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence.
is defined with some real function ξ. Assume that g, h ∈ C 1 (H), ξ ∈ C 2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξh = g has no real solution in the interior of H. Then F is uniquely determined by the functions g, h and ξ , particularly where the function s is a solution of the differential equation Remark 2.1.1. The goal of Theorem 2.1.1 is for ξ (x) to have a simple form.
Proposition 2.1.1. Let X : Ω → (0, ∞) be a continuous random variable and let , The random variable X has pd f (2) if and only if the function ξ defined in Theorem 2.1.1 has the form Proof. Let X be a random variable with pd f (2), then and finally Conversely, if ξ is given as above, then and hence Now, in view of Theorem 2.1.1, X has density (2).
Corollary 2.1.1. Let X : Ω → (0, ∞) be a continuous random variable and let h (x) be as in Proposition 2.1.1. The pdf of X is (2) if and only if there exist functions g and ξ defined in Theorem 2.1.1 satisfying the differential equation The general solution of the differential equation in Corollary 2.1.1 is where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 2.1.1 with D = 0. However, it should be also noted that there are other triplets (h, g, ξ) satisfying the conditions of Theorem 2.1.1.

Characterization Based on Hazard Function
It is known that the hazard function, h F , of a twice differentiable distribution function, F, satisfies the first order differential equation For many univariate continuous distributions, this is the only characterization available in terms of the hazard function.
The following proposition establishes a characterization of EGPL distribution, for b = 1, in terms of the hazard function, which is not of the above trivial form.
Proposition 2.2.1. Let X : Ω → (0, ∞) be a continuous random variable. The pd f of X is (2) , for b = 1, if and only if its hazard function h F (x) satisfies the differential equation with the initial condition h F (0) = 0 for β > 1.
Proof. If X has pd f (2), then clearly the above differential equation holds. Now, if the differential equation holds, then which is the hazard function of the EGPL distribution for b = 1.

Characterization Based on Reverse Hazard Function
The reverse hazard function, r F , of a twice differentiable distribution function, F, is defined as International Journal of Statistics and Probability Vol. 9, No. 3;2020 In this subsection we present a characterization of EGPL distribution in terms of the reverse hazard function.
Proposition 2.3.1. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (2) if and only if its reverse hazard function r F (x) satisfies the following differential equation Proof. If X has pdf (2), then clearly the above differential equation holds. If the differential equation holds, then from which we arrive at the reverse hazard function corresponding to the pd f (2).

Characterizations Based on Conditional Expectation
The following propositions have already appeared in (Hamedani, 2013), so we will just state them here which can be used to characterize the EGPL distribution.
Proposition 2.4.1. Let X : Ω → (c, d) be a continuous random variable with cd f F. Let ψ (x) be a differentiable function on (c, d) with lim x→c + ψ (x) = 1. Then for δ 1 , if and only if Proposition 2.4.2. Let X : Ω → (c, d) be a continuous random variable with cd f F. Let ψ 1 (x) be a differentiable function on (c, d) with lim x→d − ψ 1 (x) = 1. Then for δ 1 1 , implies that e −λx β ] a and δ 1 = b b+1 , Proposition 2.4.2 provides a characterization of EGPL distribution. (C) Of course there are other suitable functions than the ones we mentioned above, which are chosen for the sake of simplicity.

Characterizations of GZTP Distribution
We present our characterizations of GZTP via the following two Propositions.
Proposition 3.1. Let X : Ω → N be a random variable. The pm f of X is (4) if and only if Proof. If X has pm f (4), then the left-hand side of (5) will be International Journal of Statistics and Probability Vol. 9, No. 3;2020 Conversely, if (5) holds, then From (6), we also have Now, subtracting (7) from (6), we arrive at From the last equality, we have which, is the hazard function corresponding to the pm f (4).
Proposition 3.2. Let X : Ω → N be a random variable. The pm f of X is (4) if and only if its reverse hazard function, r F , satisfies the following difference equation with the initial condition r F (0) = 1.
Proof. Clearly, if X has pm f (4), then (8) holds. Now, if (8) holds, then or, in view of the initial condition r F (0) = 1, we have which is the reverse hazard function corresponding to the pm f (4)