Concomitants of Generalized Order Statistics for a Bivariate Weibull Distribution

In this paper we have studied the distribution of r–th concomitant and joint distribution of r–th and s–th concomitant of generalized order statistics for a bivariate Weibull distribution. We have derived the expression for single and product moments. Numerical study has also been conducted to see the behavior of mean of concomitants for selected values of the parameters.


Introduction
. The distribution in that case is given as , ; 0, 0, 0, 0, 0. f x y x y exp x y x y The generalized order statistics (gos) has been defined by Kamps (1995) as a unified model for ordered random variables.Kamps (1995) has argued that the quantities : , , r n m k X are called gos if their joint distribution is given as where n is sample size, m and k are parameters of the model and quantities j  are given as    The density function of a single gos is given by Kamps (1995) as ; 1,2, , Kamps (1995) has further shown that the joint density function of two GOS : , , The density functions of GOS given in (1.4) and (1.5) provide several models of ordered random variables as special case.Specifically for 0 m  and 1 k  the model reduces to Ordinary Order Statistics as given by David and Nagaraja (2003).Also for 1 m  we obtain kth upper record values introduced by Chandler (1952).Other models like fractional order statistics given by Stigler (1977), sequential order statistics etc. can also be obtained for various values of the parameters involved.Other special cases of gos can be seen in Shahbaz et al. (2017).
Sometime it happen that a sample is available from a bivariate distribution, say   , F x y , and sample is arranged with respect to one of the variable, say X.The other variable, Y, is shuffled alongside the variable X and is called the concomitant of X.When sample is arranged using order statistics then we have concomitants of order statistics and is discussed in David and Nagaraja (2003).Ahsanullah (1995) has discussed concomitants of record values.The concomitants of gos has been discussed by Ahsanullah and Nevzorov (2001) and by Shahbaz et al. (2017).The density function of rth concomitant of gos is given as where  

4). The joint distribution of two concomitants is given as
, Various authors have studied concomitants of gos.Concomitants of gos for Gumbel Bivariate Exponential distribution has been studied by Ahsanullah and Beg (2006).Further Beg and Ahsanullah (2008) has studied concomitants of GOS for Gumbel bivariate family of distributions.Nayabuddin (2013) has studied concomitants of GOS for bivariate Lomax distribution.Hanif Shahbaz and Shahbaz (2016) have studied the concomitants of gos for a bivariate exponential distribution.
In this paper we have obtained the distribution of the concomitants of upper record statistics for Bivariate Pseudo-Weibull distribution.Firstly, we have defined the Bivariate Pseudo-Weibull distribution in the following section.

Bivariate Pseudo-Weibull Distribution
The The conditional distribution of Y given The marginal and conditional distributions are useful in studying the distribution of concomitants of gos for bivariate Weibull distribution.
In the following section the distribution of concomitant of record statistics has been derived for (1.2).

Distribution of r-th Concomitant and its Properties
The Bivariate Pseudo-Weibull distribution has been given in (1.1) and (1.2).In this section the distribution of r-th concomitants of gos for Bivariate Pseudo-Weibull distribution, given in (1.2), has been obtained.
In order to obtain the distribution of concomitant of gos we first need the distribution of rth gos for the marginal distribution of X given in (2.1).The distribution of gos for X can be obtained by using (1.4).For this we first see that Now, using (2.1) and (3.1) in (1.4), the distribution of rth gos for X is where .
The conditional distribution of Y given X is given in (2.2).Now using (2.2) and (3.2) (1.6), the distribution of rth concomitant of gos for bivariate Weibull distribution is The distribution of concomitants for special cases when sample is available from a bivariate Weibull distribution can be obtained from (3.3) by using specific values of the parameters involved.
The r-th moment of the distribution given in (3.3) is obtained as: which exist for 2 p   .We can see that the moment expression given in (3.5) reduces to expression for moments of concomitants of order statistics given by Shahbaz et al. (2009) for m=0 and k=1.The table of means for m = 2, k = 2 and for various values of n,  and 2  is given below.The mean and the variance of the concomitants of gos for other values of parameters can also be tabulated.
The distribution function of rth concomitant of gos for bivariate Weibull distribution is given as The hazard rate function for concomitant of gos for bivariate Weibull distribution can be easily written by using (3.3) and (3.5) as The hazard rate function can be computed for given values of the parameters involved.
In the following section we have obtained the joint distribution of two concomitants of gos for bivariate Weibull distribution.

Joint Distribution of the Concomitants and Moments
In this section we have derived the joint distribution of the concomitants of gos for bivariate Weibull distribution given in (1.2).The joint distribution is obtained by using expression (1.7).In order to obtain the joint distribution we first obtain the joint distribution of two gos by using (1.5) and is given as where Using the distribution (4.1) and (2.2) in (1.5), the joint distribution of two concomitants of gos for bivariate Weibull distribution is x y w t    and simplifying we have Simplifying, the joint density function of two concomitants of gos for bivariate Weibull distribution is The product moments can be numerically computed from (4.3).

Conclusions and Recommendations
In this paper we have studied the distribution of concomitants of generalized order statistics when a sample is available from a bivariate Weibull distribution.The study has been conducted when   1 xx    .We have obtained the distribution of single concomitant and joint distribution of two concomitants.We have seen that the distribution of single concomitant of gos for bivariate Weibull distribution is weighted sum of Burr XII distributions.We have also seen that the mean of concomitants of gos increase with increase in the value of r until a specific point and then starts decreasing.This study can be extended by using some other choices of   x  .

A
bivariate Weibull distribution is defined by Hanif Shahbaz and Ahmad (2009) and by Ahsanullah et al. (2010) as a compound distribution of two Weibull random variables.The density function of bivariate Weibull distribution defined by Hanif Shahbaz and Ahmad (2009) is is any positive real function of X. Ahsanullah et al. (2010) have studied the distribution (1.1) for   1 xx   

( 1 . 2 )
Distribution (1.1) can be studied for other choices of   x  .The distribution (1.2) has been study in context of order statistics and record values by Ahsanullah et al. (2010) and in context of order statistics by Hanif Shahbaz et al. (2011).
bivariate pseudo Weibull distribution has been defined by Hanif Shahbaz and Ahmad (2009) as compound distribution of Weibull random variables.The density function of bivariate Weibull distribution is given in (1.2).From the density function we can readily see that the marginal density function of X is