Abstract:The concept of lower generalized order statistics was given by Pawlas and Szynal (2001). Later Burkschat et al. (2003) introduced it as dual generalized order statistics to enables a common approach to descending ordered random variables like reversed ordered order statistics, lower records and lower Pfeifer records. With this definition we shall give explicit expressions for single and product moments of lower generalized order statistics from exponentiated Weibull distribution. Further, some of its important deductions and particular cases are discussed and a characterizing result of exponentiated Weibull distribution have been obtained using conditional moments of lower generalized order statistics.

Key words:Lower generalized order statistics, order statistics, lower record values, single and product moments, exponentiated Weibull distribution and characterization.

Kamps (1995) introduced the concept of generalized order statistics. It is know that order statistics, upper record values and sequential order statistics are special cases of. In this paper we will consider the lower generalized order statistics which is given as

Let , , , be the parameters such that

, for all .

Then are called from an absolutely continuous distribution function with probability density function if their joint has the form

(1.1)

for

The marginal of th , , is

. (1.2)

The joint of and , , is expressed from (1.1) as

, , (1.3)

where

,

and

, .

We shall also take . If , , then reduces to the th order statistic, from the sample and when , then reduces to the th lower record value [Pawlas and Szynal, 2001].

Various developments on lower generalized order statistics and related topics have been studied by Pawlas and Szynal (2001), Ahsanullah (2004, 2005), Mbah and Ahsanullah (2007), Khan et al. (2008), Ather et al. (2010). Khan and Kumar (2010, 2011) and references therein.

In this study, we shall derive the explicit expressions for single and product moments of lower generalized order statistics from the exponentiated Weibull distribution. Further its various deductions and particular cases are discussed and a characterizing result of this distribution is stated and proved.

A random variable is said to have exponentiated Weibull distribution if its is given by

, , , . (1.4)

and the corresponding is

, , , . (1.5)

For more details on this distribution and its applications one may refer to Mudholkar and Srivastava (1993), Mudholkar et al. (1995), Mudholkar and Hutson (1996), Jiang and Murthy (1999) and Nassar and Eissa (2003).

Single Moments

We shall first establish some basic results which may be helpful in proving the main result.

Lemma 2.1: For the exponentiated Weibull distribution as given in (1.5) and any non-negative finite integers and ,

, (2.1)

, , (2.2)

where

. (2.3)

Proof: On expanding binomially in (2.3), we get when

, (2.4)

where

.

By setting in (2.4), we obtain

.

We have [Balakrishnan and Cohen (1991), p - 44]

, , (2.5)

where is the coefficient of in the expansion .

Therefore,

and hence the result given in (2.1).

When that

as .

Since (2.1) is form at , therefore, we have

, (2.6)

where

.

Differentiating numerator and denominator of (2.6) times with respect to , we get

.

Upon applying L’ Hospital rule, we have

. (2.7)

But for all integers and for all real numbers , we have Ruiz (1996)

. (2.8)

Therefore,

. (2.9)

Now on substituting (2.9) in (2.7), we have the result given in (2.2).

Theorem 2.1: For exponentiated Weibull distribution as given in (1.5) and , , ,

(2.10)

, (2.11)

where is as defined in (2.3).

Proof: From (1.2), we have

and hence the result given in (2.10).

On using Lemma 2.1 in (2.10), we have the result given in (2.11).

Identity 2.1: For , , and ,

. (2.12)

Proof: At in (2.11), we have

.

Note that, if , then

, and , [see Balakrishnan and Cohen (1991)]

and hence the result given in (2.12).

Remark 2.1: Setting in (2.11), the results for single moments are deduced for generalized exponential distribution, which verify the result of Khan and Kumar (2011).

Remark 2.2: Putting , in (2.11), the explicit expression for the single moments of order statistics of the exponentiated Weibull distribution can be obtained as

.

That is

,

where

.

Remark 2.3: Putting in (2.11), we deduce the explicit expression for the single moment of lower record values for exponentiated Weibull distribution in view of (2.10) and (2.2) in the form

and hence for lower records

.

Product Moments

Before coming to the main results we shall prove the following Lemmas.

Lemma 3.1: For the distribution as given in (1.5) and non-negative integers , and with ,

, (3.1)

where

. (3.2)

Proof: From (3.2), we have

, (3.3)

where

. (3.4)

Making the substitution in (3.4), we get

. (3.5)

On using (2.5) in (3.5) and simplifying the resulting expression, we obtain

.

On substituting the above expression of in (3.3), we find that

. (3.6)

Again by setting in (3.6) and then using (2.5) in resulting expression, we get the relation given in (3.1).

Lemma 3.2: For the distribution as given in (1.5) and any non-negative integers , and ,

, (3.7)

, , (3.8)

where is as given in (3.2).

Proof: When , we have

.

Now substituting for in (3.2), we get when

.

Making use of the Lemma 3.1, we derive the relation given in (3.7).

When , we have

, as .

On applying L’ Hospital rule and then using (2.9), (3.8) can be proved on the lines of (2.2).

Theorem 3.1: For exponentiated Weibull distribution as given in (1.5) and for , , ,

(3.9)

. (3.10)

Proof: From (1.3), we have

. (3.11)

On expanding binomially in (3.11), we get

and hence the result given in (3.9).

Now on making use of the lemma 3.2 in (3.9), we derive the relation given in (3.10).

Identity 3.1: For , , , and ,

. (3.12)

Proof: At in (3.10), we have

.

Note that, if , then

, , and

, , . [see Balakrishnan and Cohen (1991)]

Therefore,

.

Now on using (2.12), we get the result given in (3.12).

At , (3.12) reduces to (2.12).

Remark 3.1: Putting in (3.10), the results for product moments of Khan and Kumar (2.11) for generalized exponential distribution are deduced.

Remark 3.2: Setting , in (3.10), the explicit expression for the product moments of order statistics of the exponentiated Weibull distribution can be obtained as

.

That is

,

where

.

Remark 3.3: Putting in (3.10), we deduce the explicit expression for the product moments of lower record values for the exponentiated Weibull distribution in view of (3.9) and (3.8) in the form

and hence for lower records

.

Remark 3.4: At in (3.10), we have

,

where

, and , .

Therefore,

. (3.13)

Making use of (3.12) in (3.13) and simplifying the resulting expression, we get

as obtained in (2.11).

Characterization

Let , be from a continuous population with and , then the conditional of given , , in view of (1.3) and (1.2) is

. (4.1)

Theorem 4.1: Let be a non negative random variable having an absolutely continuous distribution function with and for all , then

, (4.2)

if and only if

, , , ,

where .

Proof: From (4.1), we have

. (4.3)

By setting from (1.5) in (4.3), we find that

, (4.4)

where

.

Again by setting in (4.4), we get

.

Simplifying the resulting expression, we derive the relation in (4.2).

To prove sufficient part, we have from (4.1) and (4.2)

, (4.5)

where

.

Differentiating (4.5) both sides with respect to , we get

or

,

where

and

.

Therefore,

which proves that

, , , .

Acknowledgments

The authors are thankful to the referee and the editor for their valuable suggestions to improve the quality of this paper.

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