A Study on Moments of Dual Generalized Order Statistics from Exponentiated Generalized Class of Distributions

In this paper, relations between moments of dual generalized order statistics from a exponentiated generalized class of distributions, given by Cardeiro . et al (2013) are studied. Some particular cases of dual generalized order statistics and examples based on it are discussed. The characterization of given distribution based on moment properties is also presented.


Introduction
The theory of ordered random variables received a tremendous attention from many researchers during the past few decades. This theory deals with the properties and applications of ordered random variables and functions involving them. There are several models of ordered random variables such as order statistics, record values, sequential order statistics, progressively Type-II censored order statistics with interesting applications in many fields of statistical science.
The concept of generalized order statistics is given by Kamps (1995), which unifies the above given models of ordered random variables arranged in ascending order as special case with various choices of the parameters involved. In real life problems sometimes, it happens that the sample is arranged in descending order for example the life length of an electric bulb arranged from highest to lowest. In such situations the distributional properties of variables cannot be studied by using the models of ordered random variables, given by Kamps (1995). The study of distributional properties of such random variables is studied by using the inverse image of generalized order statistics and is popularly known as dual (lower) generalized order statistics (dgos). This concept was first introduced by Pawlas and Szynal (2001) with name lower generalized order statistics and later extensively studied by Burkschat et at. (2003) with name dual generalized order statistics (dgos). The special cases of dgos are reverse order statistics and lower record values. these models have also been utilized in the characterization problems. Recurrence relations and identities are important in their own right because they express the higher order moments in terms of the lower order moments and hence make the evaluation of higher order moments easy as well as reduce the time and labour. For more details about theory and applications of ordered random variables, one may refer the book by Shahbaz et al. (2016).
The properties of dgos with reference to recurrence relations between moments and characterization for some specific as well as general class of distributions are studied by several authors in literature, see for example Pawlas and Szynal (2001), Ahsanullah (2004Ahsanullah ( , 2005, Mbah Two cases of dgos are given as: In view of (1) the pdf of th r dgos ( ) , , , d X r n m k is given as and the joint pdf of th r and th s dgos, and the joint pdf of th r and th s dgos, where, ( ) and corresponding pdf where 0   and 0   are the shape parameters and () Hxis the cdf of base distribution. In view of (6) and (7), it can be seen that Proof. For dgos Athar et al. (2008) have shown that Now on application of relation (8) in (10), we get which leads to (9).

Corollary 2.2:
Under the conditions as stated in Theorem 2.1 with 1  = , the relation between moment of dgos from Lehman type II is Further, the relation for single moment of order statistics can be obtained by replacing ( 1)

Remark 2.2:
The recurrence relation for the single moments of th k lower record value is

Exponentiated generalized Fréchet distribution
Let the base distribution is Fréchet distribution with cdf and the corresponding pdf Then the cdf of the exponentiated generalized Fréchet distribution is Now, we have Thus, in view of (9), we get () and at 1  = in (17), we get the relation between moment of dgos from Fréchet distribution

Exponentiated generalized power function distribution
Suppose the base distribution is power function distribution having cdf Now, we have The relation (21) is also obtained by Athar and Faizan (2011).
A Study on Moments of Dual Generalized Order Statistics from Exponentiated Generalized Class of Distributions 537

Exponentiated generalized Pareto distribution
Let the parent distribution is Pareto distribution with cdf and the corresponding pdf Thus, the cdf of the exponentiated generalized Pareto distribution is given by Now, Thus, in view of (11), we get the relation for moment of dgos from exponentiated Pareto distribution as The recursive relation for moment of dgos from exponentiated Pareto distribution with different form of pdf is also obtained by Khan and Kumar (2010).
Further at 1  = in (23), we get the relation for moment of dgos from Pareto distribution.

Exponentiated generalized Weibull distribution
Let the parent distribution is Weibull distribution distribution having cdf Thus, in view of (11), we get the relation for moment of dgos from exponentiated Weibull distribution as Similar result is also obtained by Khan et al. (2008).
Further, at 1  = in (25), we get the relation for moment of dgos for Weibull distribution. Similarly, several recurrence relations for single moment of dgos from exponentiated generalized distributions for the given parent distributions can be established using Theorem 2.1.

Exponentiated generalized Fréchet distribution
For the cdf of exponentiated generalized Fréchet distribution as given in (16) (32) Further, at 1  = in (32), we get the relation between product moment of dgos from Fréchet distribution.

Exponentiated generalized power function distribution
Let the cdf of exponentiated generalized power function distribution is as given in (19). Then, we have ( ) ( 1) ( , ) Thus, in view of (29), we get the relation for product moment of dgos from exponentiated power function distribution whereas with 1  == , we get the relation for power function distribution.
A Study on Moments of Dual Generalized Order Statistics from Exponentiated Generalized Class of Distributions 541

Exponentiated generalized Pareto distribution
For exponentiated generalized Pareto distribution as given in (22) Further, at 1  = and in view of (29), we get the relation for product moment of dgos for exponentiated Pareto distribution. Similar result is also obtained by Khan and Kumar (2010) with different form of pdf, whereas at 1  == , we have the relation for product moment of dgos from Pareto distribution.

Exponentiated generalized Weibull distribution
Let the cdf of exponentiated generalized Weibull distribution has the form as given in (24).
Therefore, we have Thus, in view of (29), we get the relation for product moment of dgos from exponentiated Weibull distribution as given by Khan et al. (2008) and when 1  == , we get the relation for product moments of dgos from Weibull distribution.
Similarly, several recurrence relations for product moment of dgos from exponentiated generalized distributions for given parent distributions can be established using Theorem 3.1.

Conclusions
Burkschat et al. (2003) proposed a model that enables a common approach to descending ordered random variables like reverse order statistics, lower record values etc. and named it dgos. Several authors studied moment properties of dgos for some specific distributions. Main aim of this paper to present a unified approach to study the moment properties of exponentiated distributions and their generalized forms by considering exponentiated generalized class of distributions as given by Cardeiro . et al (2013). This exponentiated generalized family gives greater flexibility of its tails and can be applied in many areas of engineering and biology. It is also evident that recurrence relations between moments reduce the amount of direct computation and can be used in characterization problems. This paper may be useful for those who work in the field of ordered random variable and distribution theory.