Characterizations of Kumaraswamy-Laplace , McDonald Inverse Weibull and New Generalized Exponential Distributions

Nassar (2016) considers an interesting univariate continuous distribution called Kumaraswamy-Laplace which has different forms on two subintervals. He studies certain properties and applications of this distribution. Shahbaz et al. (2016) consider another interesting distribution called McDonald Inverse Weibull distribution. They present some basic properties of their distribution and study the estimations of the parameters as well as discussing its application via an illustrative example. What is lacking in both papers, in our opinion, is the characterizations of these two interesting distributions. The present work is intended to complete, in some way, the works of Nassar and Shahbaz et al. via establishing certain characterizations of these distributions in four directions. We also introduce several New Generalized Exponential distributions and present their characterizations as well.


Introduction
The problem of characterizing a distribution is an important problem which can help the investigator to see if their model is the correct one.This work deals with various characterizations of Kumaraswamy-Laplace (KL) and McDonald Inverse Weibull (MIW) distributions to complement the works of Nassar (2016) and Shahbaz et al. (2016).These characterizations are presented in three directions: () based on the ratio of two truncated moments; () in terms of the reverse hazard function and () based on the conditional expectation of certain functions of the random variable.Similar characterizations as well as () in terms of the hazard function, will be established for several proposed New Generalized Exponential (NGE) distributions.It should be noted that characterization () can be employed also when the  (cumulative distribution function) does not have a closed form as is the case with MIW distribution.Nassar (2016)  where , ,  all positive and  ∈ ℝ are parameters.  .
The corresponding  is given by

Characterizations based on two truncated moments
This subsection deals with the characterizations of KL, MIW and NGE1-NGE5 distributions based on the ratio of two truncated moments.Our first characterization employs a theorem of Glänzel (1987), see Theorem 1 of Appendix A .The result, however, holds also when the interval  is not closed since the condition of Theorem 1 is on the interior of .
Proposition 1.Let : Ω → ℝ be a continuous random variable and let Then, the random variable  has  (2) if and only if the function  defined in Theorem 1 is of the form Proof.First, we observe that the functions , ℎ defined above are in  1 () and  is in  2 (), as required by Theorem 1 (see Appendix B).Now, suppose the random variable  has (2), then after some manipulations, we arrive at and and hence () has the above form.Further, which is clearly not equal to zero for any  <  or  ≥ .
Conversely, if  is of the above form, then Note that for = , we have . Now, according to Theorem 1,  has density (2).Corollary 1.Let : Ω → ℝ be a continuous random variable and let ℎ() be as in Proposition 1.The random variable  has  (2) if and only if there exist functions  and  defined in Theorem 1 satisfying the following differential equation The general solution of the above differential equation is where  1 and  2 are constants.We like to point out that one set of functions satisfying the above differential equation is given in Proposition 1 with  1 =  2 = Proof.Suppose the random variable  has (4), then and Further, Conversely, if  is of the above form, then from which we have () = − log {1 −  − − } ,  > 0.
Corollary 2. Let : Ω → (0, ∞) be a continuous random variable and let ℎ() be as in Proposition 2. The random variable  has  (4) if and only if there exist functions  and  defined in Theorem 1 satisfying the following differential equation The general solution of the above differential equation is where  is a constant.One set of functions satisfying the above differential equation is given in Proposition 2 with  = 0. Proof.Let  be a random variable with (6), then and finally Conversely, if  is given as above, then and hence Now, according to Theorem 1,  has density (6).
Corollary 3. Let : Ω → (0, ∞) be a continuous random variable and let ℎ() be as in Proposition 3.Then,  has  (6) if and only if there exist functions  and  defined in Theorem 1 satisfying the differential equation ,  > 0.
The general solution of the differential equation in Corollary 3 is where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 3 with  = Proof.Let  be a random variable with  (14), then and and finally Conversely, if  is given as above, then and hence Now, in view of Theorem 1,  has density (14).

Corollary 4.
Let : Ω → (0, ∞) be a continuous random variable and let ℎ() be as in Proposition 4.Then,  has  (14) if and only if there exist functions  and  defined in Theorem 1 satisfying the differential equation The general solution of the differential equation in Corollary 4 is where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 4 with  = 0.

Characterization in terms of the reverse hazard function
The

Characterization based on the conditional expectation of certain functions of the random variable
In this subsection we employ a single function  of  and characterize the distribution of  in terms of the truncated moment of ().The following proposition has already appeared in Hamedani's previous work (2013), so we will just state it as a proposition here, which can be used to characterize KL , NGE1 distributions., Proposition 7 provides a characterization of NGE1.

Characterization based on hazard function
It is known that the hazard function, ℎ  , of a twice differentiable distribution function, , 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 characterizations establish a non-trivial characterizations for (6) and (14) in terms of the hazard function which is not of the trivial form given above.Again, similar results hold for (8), ( 10) and (12).
reverse hazard function,   , of a twice differentiable distribution function,  , is defined as Let : Ω → ℝ be a continuous random variable.For  = 1, the random variable  has  (2) if and only if its reverse hazard function   () satisfies the following differential equation