On Characterizations of McLLoG, ELLoGW, PTHL and k-GE Distributions

Huang S. and Oluyede (2016), Oluyede et al. (2016), Krishnarani (2016) and Rather and Rather (2017) consider the "McDonald Log-Logistic", the "Exponentiated Log-Logistic Weibull", the "Power Transformation Half-Logistic" and "k-Generalized Exponential" distributions, respectively


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 the McDonald Log-Logistic (McLLoG), the Exponentiated Log-Logistic Weibull (ELLoWG), the Power Transformation Half-Logistic (PTHL) and the k-Generalized Exponential (k-GE) distributions to complete, in some way, the above mentioned works.These characterizations are based on: () a simple relationship between two truncated moments; () the hazard function; () reverse hazard function and () conditional expectation of a function of the random variable.It should be mentioned that for characterization () the  (cumulative distribution function) is not required to have a closed form.
where , , , ,  are all positive parameters.(2017) earlier and therefore there is no need to characterize its special case here.

Characterizations based on two truncated moments
In this subsection we present characterizations of McLLoG, ElloGW and PTHL distributions 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  is not closed.Moreover, as mentioned above, it could be also applied when the   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 (Ω, ℱ, ) be a given probability space and let  = [, ] be an interval for some  <  ( = −∞,  = ∞ mightaswellbeallowed).Let : Ω →  be a continuous random variable with the distribution function  and let  and ℎ be two real functions defined on  such that is defined with some real function .Assume that , ℎ ∈  1 (),  ∈  2 () and  is twice continuously differentiable and strictly monotone function on the set . Finally, assume that the equation ℎ =  has no real solution in the interior of .Then  is uniquely determined by the functions , ℎ and  , particularly Proof.Let  be a random variable with  (1.2), then and and finally Conversely, if  is given as above, then and hence The general solution of the differential equation in Corollary 2.1.1 is where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 2.1.1 with  = Proof.Let  be a random variable with  (1.4), then and and finally Conversely, if  is given as above, then where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 2.1.2with  = Conversely, if  is given as above, then The general solution of the differential in Corollary 2.1.3is where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 2.1.3with  = 0.However, it should be also noted that there are other triplets (ℎ, , ) satisfying the conditions of Theorem 2.1.1.

Characterization based on hazard function
It Proof.Is similar to the proof of Proposition 2.2.1 and hence omitted.

Characterization in terms of the reverse hazard function
The reverse hazard function,   , of a twice differentiable distribution function,  , is defined as  Proof.Is similar to the proof of Proposition 2.2.1 and hence omitted.

Remark 1 . 1 .
) where , ,  are positive parameters.The  (1.7) is a special case of the distribution proposed by Al-babtain et al. (2017).We believe that Rather and Rather were not aware of the work by Albabtain et al. (2017).The present author has characterized the  of Al-babtain et al.
in view of Theorem 2.1.1, has density (1.6).Let : Ω → (0, ∞) be a continuous random variable and let ℎ() be as in Proposition 2.1.3.The pdf of  is (1.6) if and only if there exist functions  and  defined in Theorem 2.1.1 satisfying the differential equation is known that the hazard function, ℎ  , of a twice differentiable distribution function, , For many univariate continuous distributions, this is the only characterization available in terms of the hazard function.The following characterizations establish non-trivial characterizations of McLLoG distribution, for  =  , ELLoGW distribution, for  = 0,  = 1 and PTHL in terms of the hazard function, which are not of the above trivial form.Proof.Is similar to the proof of Proposition 2.2.1 and hence omitted.
Proposition 2.2.3.Let : Ω → (0, ∞) be a continuous random variable.The  of  is (1.6) if and only if its hazard function ℎ  () satisfies the differential equation Let : Ω → (0, ∞) be a continuous random variable.The pdf of  is (1.2), for  = 1, if and only if its reverse hazard function   () satisfies the differential equation Proof.Is similar to the proof of Proposition 2.2.1 and hence omitted.