On Characterizations of Four Recently Introduced Distributions: Two Continuous and Two Discrete

Oluyede et al. (2016) and Mdlongwa et al. (2017) consider the continuous univariate distributions called Dagum-Poisson (DP) and Burr XII Modified Weibull (BXIIMW), respectively, and study certain properties and applications of these distributions. Shahid and Raheel (2019) and Para and Jan (2019) proposed the univariate discrete distributions called Discrete Modified Inverse Rayleigh (DMIR) and Discrete Generalized Inverse Weibull (DGIW) and study some of their mathematical properties. The present short note is intended to complete, in some way, the works cited above via establishing certain characterizations of these 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 DP, BXIIMW, DMIR and DGIW distributions to complete, in some way, the works cited above. These characterizations are based on: (i) a simple relationship between two truncated moments; (ii) the hazard function and (iii) conditional expectation of a function of the random variable, for DP and BXIIMW. It should be mentioned that for characterization (i) the cdf (cumulative distribution function) is not required to have a closed form. As for the DMIR and DGIW, certain characterizations of these distributions are presented based on: different form of (iii) which we call it (iii′) and (iv) the reverse hazard function. Oluyede et al. (2016) introduced the DP distribution with cdf and pdf (probability density function) given, respectively, by where        are all positive parameters. Mdlongwa et al. (2017) proposed the BXIIMW distribution with cdf, pdf and hazard function given, respectively, by , , and   where   are positive parameters and is the set of all positive integers.
where   are positive parameters.

Characterizations of DP and BXIIMW Distributions
We present our characterizations (i)-(iii) in three subsections.

Characterizations Based on Two Truncated Moments
In this subsection we present characterizations of DP and BXIIMW distributions in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glä nzel (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 cdf F does not have a closed form. As shown in Glä nzel (1990), this characterization is stable in the sense of weak convergence.
and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation hg   has no real solution in the interior of H. Then F is uniquely determined by the functions , gh and , The general solution of the differential equation in Corollary 2.1.1 is where D is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Now, in view of Theorem 2.1.1, X has density (4).
The general solution of the differential equation in Corollary 2.1.2 is Note that a set of functions satisfying the above differential equation is given in Proposition 2.1.2 with D = 0. However, it should be also noted that there are other triplets   ,, hg satisfying the conditions of Theorem 2.1.1.

Characterization Based on Hazard Function
It is known that the hazard function, F h , 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 BXIIMW distribution in terms of the hazard function, which is not of the above trivial form.

Proposition 2.2.1. Let X:
  0,    be a continuous random variable. The pdf of X is (4) if and only if its hazard function   F hx satisfies the differential equation with the initial condition   0 0 F h  for c>0 and    Proof. If X has pdf (4), then clearly the above differential equation holds. Now, if the differential equation holds, then International Journal of Statistics and Probability Vol. 8, No. 4;2019 29 which is the hazard function of the BXIIMW distribution.

Characterizations Based on Conditional Expectation
The following proposition has already appeared in Hamedani (2013), so we will just state it here which can be used to characterize the BXIIMW distribution.