Some Extended Classes of Distributions: Characterizations and Properties

Based on a simple relationship between two truncated moments and certain functions of the nth order statistic, we characterize some extended classes of distributions recently proposed in the statistical literature, videlicet Beta-G, Gamma-G, Kumaraswamy-G and McDonald-G. Several properties of these extended classes and some special cases are discussed. We compare these classes in terms of goodness-offit criteria using some baseline distributions by means of two real data sets.


Introduction
The recent literature has suggested several ways of extending well-known distributions.One of the earliest is the class of distributions generated by a standard beta distribution pionnered by Eugene et al. (2002).The more recent ones are: the class of distributions generated by Kumaraswamy (1980)'s distribution defined by Cordeiro and de Castro (2011), and the class of distributions generated by McDonald (1984)'s generalized beta distribution introduced by Alexander et al. (2012).Generalized distributions usually provide flexible framework for modeling a wide range of data sets, that is, these models are very useful for fitting a wide spectrum of real world lifetime data in biology, medicine, engineering, economics and other fields.
For example, the pdf and cdf of the Kumaraswamy-inverse Weibull (Kw-IW) distribution are given by respectively, where  > 0,  > 0,  > 0 and  > 0 are parameters.
The corresponding cdf is given by where  > 0 and  > 0 are shape parameters,  > 0 is a scale parameter and  denotes the vector of unknown parameters in ().We assume that () is a monotonically increasing function of  with () ≥ 0, lim →0 + () = 0 and the derivative () = d()/d belongs to the interval (0, ∞).A characterization of the BEW family is that its hazard rate function (hrf) can be bathtub shaped, monotonically increasing or decreasing and upside-down bathtub depending basically on the parameter values.This family contains as special models several well-known distributions.Some useful distributions in this family are presented in Cordeiro et al. (2012).
The generator proposed by Zografos and Balakrishnan (2009) and Ristic′ and Balakrishnan (2012), called the gamma-G ("GG" for short) family defined from any baseline cdf (;  ),  ∈ ℝ, considers an extra shape parameter  > 0. They defined the GG family by the pdf and cdf and   (;  , ) = respectively, where  > 0 is a shape parameter,  > 0 is a scale parameter and  is a vector of unknown parameters in ().We assume that () ≥ 0 is monotonically increasing in  with lim →0 + () = 0, lim →∞ () = ∞ and the derivative () = ()/ is defined in (0, ∞).The proposed family includes several well-known models as special cases such as the exponential, Pareto, Gomertz, Weibull and modified Weibull distributions, among others.
The distribution proposed by Mead (2014) The goal of this paper is to provide characterizations of the McG-K, BEW, GEW, McN and Kw-IW families described above.These characterizations are based on: () a simple relationship between two truncated moments, () certain functions of the th order statistic, () certain functions of the first order statistic.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 (1978), Glänzel (1987), Hamedani (1993Hamedani ( , 2002Hamedani ( , 2006)) These classes of distributions provide tools to obtain new parametric distributions from existing ones and have applications in many fields, in particular in lifetime modeling.
The paper is organized as follows.In Section 2, we consider a characterization based on two truncated moments.In Section 3, we discuss about characterizations based on truncated moment of the ℎ order statistic.In Section 4, we provide characterizations based on truncated moment of the first order statistic.In Section 5, we derive expansions for the pdfs of those families as linear combinations of exponentiated -G (Exp-G) families, where G is the baseline model.Some mathematical properties are addressed (Section 6) and two applications are explored to prove the efficiency of the new generators (Section 7).Some concluding remarks are provided in Section 8.

Characterization based on two truncated moments
In this section, we present characterizations of the McG-K, BEW, GEW, McN and Kw-IW families in terms of a simple relationship between two truncated moments.The characterizations derived here employ an interesting result due to Glänzel (1987), which is given by the following theorem.
Proof.Let  have pdf (1).Then, for  > 0, Observe that, Conversely, if  is given as above, then for  > 0 , where  is a constant.One set of appropriate functions satisfying the above equation is given in Proposition 1.2 with  = 0.  where  = 0.

Truncated moment of the 𝒏𝒕𝒉 order statistic
Let  1: ≤  2: ≤. . .≤  : be the corresponding order statistics from a random sample of size  from a continuous  .We briefly discuss here characterization results based on functions of the ℎ order statistic.We have the following proposition.
Proof.If (11) holds, then using integration by parts on the left hand side of ( 11) and the condition lim →0 ()()  = 0 , we have ∫

Characterizations based on the truncated moment of the first order statistic
We state here two characterizations based on certain functions of the first order statistic.We like to mention that the proof of Proposition 4.1 below is a straightforward extension of Theorem 2.2 of Hamedani (2010).We give a short proof of it for the sake of completeness.} ,  ≥ 0.
Proof.If (13) holds, then using integration by parts on the left hand side of ( 13) and the assumption lim →∞ ()[1 − ()]  = 0, we have Differentiating both sides of the above equation with respect to , we obtain ,  > 0.
Now, integrating both sides of ( 14) from 0 to , we have, in view of ∫

Useful representation
Theorem 5.1 Let  be a random variable having any of the five families of distributions discussed so far and the function   (, ) = ( + ), where  = 1,2, … and ,  ∈ ℝ + .The pdf of  can be expressed as the linear combination where ℎ (+) () denotes the  −  (( + )) density function.

Mathematical properties
In this section, we derive moments, moment generating function (mgf) and quantile function (qf) of those distributions.

Moments
We derive several representations for the moment   ′ = (  ) of  having all of five families discussed in this paper.Note that other kinds of moments related to the Lmoments of Hosking (1990) may also be obtained in closed-form, but we confine ourselves here to   ′ for brevity.
Henceforth, we assume that  (+) ~Exp-G(( + )).The importance of moments in Statistics especially in applications is obvious.A first formula for the th moment of  can be obtained from (15) and the monotone convergence theorem as   ′ = (  ) = ∑ ∞ =0   ( (+)

Moment generating function
The mgf provides the basis of an alternative route to analytical results compared with working directly with the pdf and cdf and it is widely used in the characterization of distributions and the application of the skew-normal test (Meintanis, 2010) and other goodness of fit tests (Ghosh, 2013).

Applications
In this section, we compare the fits of the BG, GG, KwG and McG with the baselines Gamma (Γ), Weibull (W) and Inverse Weibull (IW) to two real data sets from Murthy et al. (2004).
Table 1 provides a summary of these data.The stress data have positive skewness and negative kurtosis.  2 indicate that the Kw-G model has the smallest values of these statistics among all fitted models.So, it could be chosen as the more suitable model in these cases (when we use gamma and IW as the parent distributions).However, note that when the baseline is Weibull, the GG family presents better performance than the others.Thus, we can say that is important to propose new generators in order to provide better fits to real data sets.Besides that, note that when we compare the models KwΓ, GW and KwIW (models that field better adjustments), the best of them was the second, showing, in this study, that the gamma generator provides the best performance among the others generators.Moreover, we also provide a visual comparison of the histogram of the data with the fitted density functions.The plots of the fitted densities for the baselines Γ, W and IW are displayed in Figures 1(a), 1(b) and 1(c), respectively, for the data set.We only reinforce what has been said above.
Figure 1: Estimated densities of the selected generators for stress data.

Application 2: Repairable data
The following data refer to the time between failures for repairable itens ( = 30): Table 3 provides a summary of these data.The repairable data has positive skewness and kurtosis, and has less variability.4 indicate that the GG model has the smallest values of these statistics among all fitted models.So, it could be chosen as the more suitable model in this case (when we take gamma and Weibull as the baselines).However, note that when the baseline is Weibull, the GG generator presents better performance than the others, as in the first application.Besides that, note too that when we compare the GΓ, GW and KwIW models (those that yield better adjustments), the best of them is the second, showing, in this study, that the GG generator provides the best performance among the other current models.These results are exhibited in Figure 2.

( 2 )
as the McDonald generalized-K (denoted by the prefix "McG-K" for short) family since the McDonald density function is a basic exemplar when () =  for  ∈ (0,1).The family of distributions (2) includes two important special classes: the beta generalized (BG) (Eugene et al., 2002) for  = 1, and the Kumaraswamy generalized (KwG) (Cordeiro and de Castro, 2011) for  = 1.It follows from (2) that the McG-K family with baseline cdf () is the BG distribution with baseline cdf ()  .This simple transformation may facilitate the derivation of some of its structural properties.For example, the pdf and cdf of the McDonald Normal (McN) distribution are given by (; , , , , )

Figure 2 :
Figure 2: Estimated densities for the selected generators for repairable data.

Table 2
lists the values of the following statistics for some models: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (AICc) and Bayesian Information Criterion (BIC).The figures involving the Γ and IW baselines in Table

Table 4
lists the values of the following statistics for some models: AIC, AICc and BIC.The figures involving Γ and  baselines in Table