Moments of Power Function Distribution Based on Ordered Random Variables and Characterization

In this paper simple expressions for single and product moments of generalized order statistics from the power function distribution have been obtained. The results for order statistics and records are deduced from the relations derived. Further, a characterizing result of this distribution on using the conditional moments of the generalized order statistics is discussed.


Introduction
The concept of generalized order statistics ) (gos was introduced by Kamps   (1995).Several models of ordered random variables such as order statistics, record values, sequential order statistics, progressive type II censored order statistics and Pfeifer's record values can be discussed as special cases of the gos .Suppose ) , , , is a real number), are n gos from an absolutely continuous distribution function , then this model reduces to the ordinary  r th order statistic and (1) will be the joint pdf of n order statistics , then (1) will be the joint pdf of the first n record values of the identically and independently distributed ) (iid random variables with df ) (x F and corresponding pdf ) (x f . In view of (1), the marginal pdf of the  r th gos, where Several authors utilized the concept of gos in their work.References may be made to Kamps and Gather (1997), Keseling (1999), Cramer and Kamps (2000), Ahsanullah (2000Ahsanullah ( , 2004)), Habibullah and Ahsanullah (2000), Pawlas and Szynal (2001), Raqab (2001), Kamps and Cramer (2001), Ahmad and Fawzy (2003), Bieniek and Szynal (2003) In this paper we have obtained simple expressions for the exact moments of generalized order statistics from the power function distribution.Results for order statistics and record values are deduced as special cases and a characterization of this distribution is obtained by using conditional moments of generalized order statistics.
A random variable X is said to have the three parameter power function distribution if its probability density function ) ( pdf is of the following form 1 ) ( This is a Pearson's Type-I distribution.If 1 


then the power function distribution coincides with the uniform distribution on the interval ) , ( We will consider in this paper without any loss of generality 0   and the corresponding distribution function ) (df For applications of the distribution one may refer to Meniconi and Barry (1996), Zaka and Akhter (2013) and Arslan (2014).

Single Moments Lemma 2.1
For the power function distribution as given in (5) and any non-negative finite integers a and b , when where ISBN-1391-4987 IASSL Proof.On expanding binomially the term where Setting the expression ( 6) is undefined, therefore we have to consider the limit as m tends to 1 where Differentiating numerator and denominator of (10) b times with respect to m , we get On applying the L' Hospital rule, we have But for all integers 0  n and for all real numbers x , from Ruiz (1996 Now on substituting ( 13) in (11), we have the result given in (7).

Theorem 2.1
For the power function distribution as given in (5) and Proof.From ( 2) and ( 8), we have Making use of Lemma 2.1, we establish the relation given in (15).
Proof.(16) can be proved by setting 0  j in (15).(15), the explicit formula for the single moments of order statistics of the power function distribution can be obtained as

Special cases
in (15), we deduce the explicit expression for the single moments of upper k record values for the power function distribution in view of ( 7) and (14)

Product Moments Lemma 3.1
For the power function distribution as given in (2) and non-negative integers a , b and Proof.From ( 20), ( 5) and ( 4), we have where IASSL ISBN-1391-4987 97 By setting Again by setting in (23) and simplifying the resulting expression, we derive the relation given in (19).

Lemma 3.2
For the condition as stated in Lemma 3.1, and ) , , ( , c b a J j i is as given in (20).
and for 20), we get . Making use of Lemma 3.1, we derive the relation given in (25).
, so after applying L'Hospital rule and ( 13), ( 25) can be proved on the lines of (7).

Theorem 3.1
For the power function distribution as given in (5) and binomially in (29), we get the relation given in (27).
Making use of Lemma 3.2, we derive the relation given in (28).
Now on using ( 16), we get the result given in (30).28), the explicit formula for the product moments of order statistics of the power function distribution can be obtained as

Special cases
ii) Letting m tends to 1  in (28), we deduce the explicit expression for the product moments of upper k record values for the power function distribution in view of ( 27) and ( 26) in the form Making use of (30) in (31) and simplifying the resulting expression, we get as obtained in (15).

Characterization
, in view of ( 2) and (3), is Proof.From (32), we have 5) in (34), we obtain where Again by setting  To prove the sufficient part, we have from (32) and (33)  , we obtain the characterization results of the power function distribution based on order statistics and record values, respectively.