Characterizations of NWP , ETGR and TWL Distributions

Utilizing a simple relationship between two truncated moments as well as certain functions of the and of the order statistics, we characterize three extended classes of distributions proposed in (2015).


Introduction
The recent literature has suggested several ways of extending well know distributions.In a general way, generalized distributions provide a flexible framework for modeling a large range of data, that is, these models provide a rather flexible mechanism for fitting a wide spectrum of real world lifetime data in biology, medicine, engineering, economics, sports and other areas.In what follows we consider three generalized families of distributions introduced in 2015.
1) Nasiru and Luguterah (2015) proposed a New Weibull-Pareto (NWP) family of distributions with probability density function (pdf) given (in their own notation) by and cumulative distribution function (cdf) in the form where are all positive parameters.
2) Afify et al. (2015a) introduced the Exponentiated Transmuted Generalized Rayleigh (ETGR) family of distributions.The pdf of the ETGR family (in their own notation) takes the form ( ) whereas the cdf is given by ( ) where all positive and | | are parameters.
3) Afify et al. (2015b) proposed a new family of distributions called the Transmuted Weibull Lomax (TWL) family of distributions.The pdf and cdf of the TWL family (in their own notation) are given, respectively, by ( ) and where , all positive and | | are parameters.
It is widely known that the problem of characterizing a distribution is an important issue which has attracted the attention of many researchers.Thus, various characterizations have been established in many different directions.For example, we can refer to Galambos and Kotz ( ), Glänzel et al.(1984), Glänzel (1987Glänzel ( , 1988Glänzel ( , 1990)), Hamedani (1993Hamedani ( , 2002Hamedani ( , 2006)) The goal of this note is to provide characterizations of the NWP, ETGR and TWL families of distributions described above.These characterizations are based on: ( ) a simple relationship between two truncated moments , ( ) certain functions of the order statistic, ( ) certain functions of the order statistic.
Although in many applications an increase in the number of parameters provides a more suitable model, in characterization problems a lower number of parameters (without seriously affecting the suitability of the model) is mathematically more appealing (see Glänzel and Hamedani 2001).In the applications where the underlying distribution is assumed to be NWP or ETGR or TWL distribution, the investigator needs to verify that the underlying distribution is in fact the NWP or ETGR or TWL distribution.To this end the investigator has to rely on the characterizations of these distributions and determine if the corresponding conditions are satisfied.Thus, the problem of characterizing these families of distributions become essential.As we mentioned earlier, our objective here is to present characterizations of the NWP, ETGR and TWL families of distributions.We shall do this in three different directions as discussed in Section 2 below.

Characterization Results
The NWP, ETGR and TWL classes of distributions provide tools to obtain new parametric distributions from existing ones and have applications in many fields of study, in particular in lifetime modeling.So, an investigator will be vitally interested to know if their model fits the requirements of NWP or ETGR or TWL distribution.To this end the investigator riles on characterizations of these distributions, which provide conditions under which the underlying distribution is indeed a NWPor ETGR or TWL distribution.
In this section we will present various characterizations of these distributions.First, we will consider characterizations based on two truncated moments.Next, characterizations based on truncated moments of certain functions of the order statistic and after that based on the order statistic.

Characterizations based on two truncated moments
In this subsection we present characterizations of the NWP, ETGR and TWL families of distributions in terms of a simple relationship between two truncated moments.The results derived here will employ an interesting theorem due to Glänzel (1987), which is given below.
Remarks 1. ( ) In Theorem G, the interval need not be closed.( ) The goal is to have the function as simple as possible.( ) It is possible to state Theorem 1 based on two functions and by setting ( ) as we intend to do in the following Proposition.