Some Results on Exponentiated Weibull Distribution via Dual Generalized Order Statistics

In this paper, we use the concept of dual generalized order statistics dgos which was given by Pawlas and Syznal (2001). By using this, we obtain the various theorems and some relations through ratio and inverse moment by using exponentiated-Weibull distribution. Cases for order statistics and lower record values are also considered. Further, we characterize the exponentiated-Weibull distribution through three different methods by using the results obtained in this paper.


Introduction
The exponentiated Weibull distribution was proposed by Mudholkar and Hutson (1996). For more distributional properties of the exponentiated Weibull distribution, we may refer to Mudholkar, Srivastava and Freimer (1995) and Nassar and Eissa (2003).
A random variable X is said to have exponentiated Weibull distribution (Mudholkar and Hutson (1996)) if its probability density function ) ( pdf is of the form ( The exponentiated Weibull distribution is the extended form of the two parameters weibull distribution. Also, it shows many characteristics quite similar to exponential; Weibull and exponentiated exponential distributions and their df and the pdf are found to have closed forms. We can apply this distribution even on censored data. Pawlas and Syznal (2001) For the case m m i = , Several authors utilized the concept of dgos in their work. References may be made to Pawlas and Szynal (2001), Ahsanullah (2004Ahsanullah ( , 2005, Mbah and Ahsanullah (2007), Khan et al. (2006, Kumar (2010, 2011) and Khan and Khan (2015) among others. In this paper, we mainly focus on the study of dgos arising from the exponentiated Weibull distribution.

Some Theorems and Useful Results
Note that for exponentiated Weibull distribution ) (x f and ) (x F satisfy the relation The relation in (6) will be used to derive some simple recurrence relations for the moments of dgos from the exponentiated Weibull distribution.

Relation for Inverse Moments
where Case I: Making the substitution (9), we find that On using the logarithmic expansion (see Balakrishnan and Cohen (1991), p-44)), Case II: where Differentiating numerator and denominator of (13) b times with respect to m , we get On applying L' Hospital rule, we have But for all integers 0  n and for all real numbers x , we have Ruiz (1996) Some Results on Exponentiated Weibull Distribution via Dual Generalized Order Statistics 214 (15) Therefore, On substituting (16) in (14), we find that Now substituting for (7) and simplifying, we obtain when (18) and , in view of (17) and (7), we have where ) (k r Z denote the − k th lower record value. We obtain the recurrence relations for single moments of exponentiated Weibull distribution.

Theorem 1.
For the distribution as given in (2) for where Proof. In view of Khan .
By using (6) and hence the result given in (21). (21), we obtain a recurrence relation for single moments of order statistics of the exponentiated Weibull distribution of the form in (21), we get a recurrence relation for single moments of lower k record values from exponentiated Weibull distribution in the form of

Relation for Ratio Moments
The explicit expressions for the product moments of On expanding where .
Case I: where By setting (27) and simplifying on the line of (12), we find that By substituting the expression of Again by setting (28) and simplifying the resulting expression, we obtain Case II: Therefore, on applying L' Hospital rule and using (16), we find that Now on substituting for ( ) from (29) in (24) and simplifying, we obtain when and , in view of (30) and (25), we have which is the exact expression for single moment as given in (18).

Theorem 2.
For the distribution as given in (2), for n s r where Proof. In view of Khan .

Special Cases of Ratio and Inverse Moments
in (18), the explicit formula for single moments of order statistics of the exponentiated Weibull distribution can be obtained as

Characterization
, in view of (4) and (5), is Theorem 3. Let X be a non-negative random variable having an absolutely continuous df if and only if , we have from (36) for 1 + r By setting  To prove sufficient part, we have from (36) and (38) Therefore, Integrating both the sides of (43) with respect to x between ) , 0 ( y , the sufficiency part is proved. , we find that On using (45) in (44), we have the result given in (39).
Sufficiency part can be proved on the lines of case 1 −  m .
Theorem 4. Let X be a non-negative random variable having an absolutely continuous df if and only if Proof. The necessary part follows immediately from (21). On the other hand if the recurrence relation in (46) is satisfied. In view of Khan . (47) Now applying a generalization of the Müntz-Szász Theorem (Hwang and Lin, 1984) to (48), we get which proves that Theorem 5. Suppose an absolutely continuous (with respect to Lebesgue measure) random variable X has the df where ) ( x F To prove sufficient part, we have from (49) ) ( (52) Differentiating (52) on both the sides with respect to x , we find that ) Therefore,

Conclusion
This paper demonstrates the explicit expressions for ratio and inverse moment for single moment as well as product moment through lower (dual) generalized order statistics. Some recurrence relations for single and product moments of dgos from the exponentiated Weibull distribution have been established. respectively. Further, conditional expectation of function, recurrence relation and truncated expectation of dgos has been utilized to obtain a characterization of the exponentiated Weibull distribution and some important results are deduced.