Moment Generating Functions of Generalized Order Statistics From Extended Type II Generalized Logistic Distribution

In this paper, explicit expressions and some recurrence relations are derived for marginal and joint moment generating functions of generalized order statistics from extended type II generalized logistic distribution. Further the results are deduced for moments of − k th record values and ordinary order statistics.


Introduction
A random variable X is said to have extended type II generalized logistic distribution if its probability density function ) ( pdf is of the form 1 ) 1 ( ) ( (1.1) and the corresponding survival function is For more details on this distribution and its application one may refer to Balakrishnan and Leung [4].
The logistic distribution plays an important role in growth curve have made it one of the many important statistical distributions. The shape of the logistic distribution that is similar to that of the normal distribution makes it simpler and also profitable on suitable occasions to replace the normal by the logistic distribution with negligible errors in the respective theories.
Kamps [6] introduced and extensively studied the generalized order statistics ) ( and n is a positive integer.
Choosing the parameters appropriately, models such as ordinary order statistics , order statistics with non-integral sample size 1 ( and progressive type II censored order are obtained (Kamps [6], Kamps and Cramer [7]).
Marshall-Olkin extended logistic and extended type I generalized logistic distribution respectively. Al-Hussaini et al. [2,3] have established recurrence relations for conditional and joint moment generating functions of gos based on mixed population, respectively. Kumar [9] have established explicit expressions and some recurrence relations for moment generating function of record values from generalized logistic distribution.
In the present study, we establish exact expressions and some recurrence relations for marginal and joint moment generating functions of gos from extended type II generalized logistic distribution. Results for order statistics and record values are deduced as special cases.

Relations for marginal moment generating function
Note that for extended type II generalized logistic distribution defined in (1.1) (2.1) The relation in (2.1) will be exploited in this paper to derive exact expressions and some recurrence relations for the moment generating function of gos from the extended type II generalized logistic distribution. We shall first establish some basic results which may be helpful in proving the main result. Lemma 2.1: For the extended type II generalized logistic distribution as given in (1.1) and any non-negative and finite integers a and b is the beta function.

Proof: On expanding
Making use of Lemma 2.1, we establish the result given in (2.6). (2.8) Using the following results

Published by Atlantis Press
Copyright: the authors 277 we can obtain the moments of any value of r .
and hence the result given in (2.11). Making use of (2.7) in (2.11), we establish the result given in (2.12 ii) Setting 1 = k in (2.12), we get the explicit expression for marginal moment generating function of upper k record values from extended type II generalized logistic distribution can be obtained as A recurrence relation for marginal moment generating function for gos from df (1.2) can be obtained in the following theorem.

Theorem 2.2:
For the distribution given in (1.1) and for (2.14) Integrating by parts treating ) for integration and rest of the integrand for differentiation, we get Differentiating both the sides of (2.15) j times with respect to t , we get

Relations for joint moment generating function
Before coming to the main results we shall prove the following Lemmas.

Published by Atlantis Press
Copyright: the authors 281 RETRACTED On substituting the above expression of (3.5) Again by setting (3.5) and simplifying the resulting expression, we derive the relation given in (3.1).
Making use of the Lemma 3.1, we established the result given in (3.7).
Published by Atlantis Press Copyright: the authors 282 On using the logarithmic expansion

(Balakrishnan and
Cohen [5], Shawky and Bakoban [18]), integrating the resulting expression we get On substituting the above expression of ) (x I in (3.9), we find that in (3.10) and simplifying the resulting expression, we derive the relation given in (3.8).
Theorem 3.1 For extended type II generalized logistic distribution as given in (1.1) and for n s (3.14) Proof binomially in (3.15) and simplifying the resulting expression, we have the result given in (3.11). Making use of (3.7) in (3.11), we establish the relation given in (3.12).

Special cases
i) Putting 0 = m , 1 = k in (3.12), the explicit formula for joint moment generating function of order statistics for the extended type II generalized logistic distribution can be obtained as Making use of (2.1), we can derive the recurrence relations for joint moment generating function of gos from (1.5).

Theorem 3.2:
For the distribution given in (1.1) and for n s