1. Introduction

A random variable X is said to have a Kumaraswamy distribution (KD) if its probability density function is (pdf) in the form:

(1.1)

The cumulative distribution func-tion (CDF) and survival function (SF) are:

(1.2)

and

(1.3)

More details on this distribu-tion and its applications can be found in Kumaraswamy ^[19], Sundar and Subbiah ^[24], Fletcher and Ponnambalam ^[10], Seifi et al. ^[23], Ganji et al. ^[11], Sanchez et al. ^[22] and Courard-Hauri ^[9].

Marshall and Olkin ^[19] intr-oduced a new method of adding a parameter into a family of distrib-utions. According to them if denote the survival or reliability function of a continuous random variable X then the timely honored device of adding a new parameter results in another survival function defined by

(1.4)

The pdf and hazard rate function (HRF) corresponding to are:

(1.5)

and

(1.6)

where, is the HRF correspo-nding .

Here, we substituting from (1.1) and (1.3) in (1.5) to obtain a new distribution called Marshall-Olkin extended Kumaraswamy distribu-tion (M-OEKD) as follows

(1.7)

The CDF, SF and HRF corresponding to are:

(1.8)

(1.9)

and

(1.10)

In view of (1.7) and (1.9), we have

(1.11)

The inverse of the distribution function (1.8) yields a very simple quantile function

(1.12)

which facilitates ready quantile based statistical modeling. In addition, gives a trivial random variable generation. If , then is given by

(1.13)

The mode for M-OEKD can be obtained as the root of the following equation:

(1.14)

and the median for M-OEKD is

(1.15)

This distribution can be applied on some real percentage data. Carrasco et al ^[16] applied Generalized Kumaraswamy Distribution on the observed percentage of children living in households with per capita income less than R$ 75.50 in 1991 in 5509 Brazilian municipal districts.

The concept of generalized order statistics (gos) was introduced by Kamps ^[13]. A variety of order models of random variables is contained in this concept.

Let, for simplicity, throughout denote an absolutely continuous distribution function with density function f. The random variables are called generalized order statistics based on, if their joint pdf of the form

for , with parameters such that for all .

For for all the pdf of is given by Kamps and Cramer ^[14] in the following way

(1.16)

The joint pdf of and , is given as

(1.17)

where and

It may be noted that for

(1.18)

and

(1.19)

Therefore pdf of given in (16) reduces to

(1.20)

and joint pdf of and given in (1.17) reduces to

(1.21)

where

and

We shall also take . If m = 0, k = 1, then reduces to the order statistics, from the sample and when m = -1, then reduces to the k^th record values (Pawlas and Szynal ^[21]).

Many authors utilized the gos in their work, such as Kamps and Gather ^[15], Athar et al ^[7], Cramer and Kamps ^[8], Ahsanullah ^[4], Pawlas and Szynal ^[21], Ahmed ^[2], Ahmed and Fawzy ^[3], Khan et al. ^[17], AL-Hussaini et al. ^[5] and Kumar ^[18]. Abdul-Moniem ^[1] obtained recurrence relations for moments of lower gos from exponentiated Lomax distribution and its characterization.

In this paper, we have established explicit expressions and some recurrence relations for single and product moments of gos from MOEKD family of life distributions. Further its various deductions and particular cases are discussed. Characterization of M-OEKD has been obtained on using a recurrence relation for single moments.

2. Single Moments of gos

Theorem 2.1. let be a random variable has pdf (7). Then for integer such that , the follow-ing recurrence relation is satisfied.

(2.1)

Proof. We have from Lemma 2.3 (Athar and Islam ^[6]) that

If we let , then

(2.2)

On using (1.11) in (2.2), we get

Which after simplification leads to (1.2).

Corollary 2.2. For the recurrence relations for single moment of gos for M-OEKD is given as

(2.3)

Proof. This can easy be deduced from (2.1) in view of the relation (1.18).

Note that: We can obtain the recurrence relations for single moment of gos for Kumaraswamy distribution by taking in (2.3), established by Kumar ^[18].

Remark 2.3 Putting in (2.3), we obtain recurrence relations for single moments of order statistics as

(2.4)

Remark 2.4 Setting in (2.3), we obtain the recurrence relations of upper K-record values as

(2.5)

3. Product Moments of gos

Theorem 3.1 let be a random variable has pdf (1.7). Then for integer such th-at , the following recurrence relation is satisfied.

(3.1)

Proof. We have from Lemma 3.2 (Athar and Islam ^[6]) that

If we let , then

In view of (1.11), note that

Therefore,

Which after simplification leads to (3.1).

Corollary 3.2. For the recurrence relations for product moments of gos for M-OEKD is given as

(3.2)

Proof. This can easy be deduced from (3.1) in view of the relation (1.19).

Note that: We can obtain the recurrence relations for product moments of gos for Kumaraswamy distribution by taking in (3.2), established by Kumar ^[18].

Remark 3.3 Putting in (3.2), we obtain recurrence relations for product moments of order statistics as

(3.3)

Remark 3.4 Setting in (3.2), we obtain the recurrence relations for product moments of k^th record values as

(3.4)

4. Characterization

Theorem 4.1 Let X be a non-negative random variable having an absolutely continuous distribution function F (x) with F (0) = 0 and 0 < F (x) < 1 for all x > 0, then

(4.1)

if and only if .

Proof The necessary part follows immediately from equation (2.3). On the other hand if the recurrence relation in equation (4.1) is satisfied, then on using equation (1.20), we have

Integrating the first integral on the right hand side of the above equation, by parts, we get

Which is implies that

(4.2)

Now applying a generalization of the Muntz-Szasz theorem (Hwang and Lin ^[12]) to equation (4.2), we get

which prove that

5. Conclusion

Here we propose a new model, the so-called the Marshall – Olkin exte-nded Kumaraswamy distribution which extends the Kumaraswamy distribution. Some properties for this distribution have been obtained. We give recurrence relations for single and product moments of generalized order statistics from Marshall – Olkin extended Kumaraswamy distribution. Specializations to order statistics and records have been made. We obtain characterization of Marshall – Olkin extended Kumaraswamy distribution using a recurrence relation for single mom-ents.

Acknowledgment

The author is thankful to the Editor and the Referee for their fruitful suggestions and comments.

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