CONCOMITANTS OF ORDER STATISTICS FOR BIVARIATE EXPONENTIATED INVERTED WEIBULL DISTRIBUTION

In this paper, bivariate exponentiated inverted Weibull distribution (BEIWD) has been proposed using the Morgenstern approach. We provide the pdf and cdf of exponentiated inverted Weibull distribution (EIWD) along with the order statistics and its important properties. We derive the Morgenstern type bivariate exponentiated inverted Weibull distribution (BEIWD) and discussed some of its properties. We obtain the concomitants of order statistics arising from BEIWD with its several distributional properties. The survival and hazard functions of concomitants of its order statistics are illustrated with graphical presentation and numerical computations for different sets of values of parameters.


INTRODUCTION
The exponentiated inverted Weibull distribution (EIWD) is a life time probability distribution for 6445 BIVARIATE EXPONENTIATED INVERTED WEIBULL DISTRIBUTION modeling reliability data which is a generalization of inverted Weibull distribution (IWD). Flaih, Elsalloukh, Mendi and Milanova [1] proposed this distribution by raising the cumulative density function (cdf) of IWD to a non-negative parameter (say , ∈ ℜ) by exponentiation. The exponentiated inverted Weibull distribution (EIWD) has been studied by many authors, for example; Aljuaid [3] estimated the parameters of EIWD under type-II censoring. Elbatal and Muhammed [9] proposed exponentiated generalized inverse Weibull (EGIW) distribution.
Hassan, Marwa, Zaher and Elsherpieny [18] studied estimation of stress-strength reliability for EIWD based on lower record values. Ahmad, Ahmad and Ahmad [2] obtained bayesian estimators of the shape parameter of the EIWD. Saghir, Tazeem and Ahmad [4] innovated a new three parameter weighted exponentiated inverted Weibull distribution (WEIWD).
A bivariate distribution ( , ) for a pair of random variables ( , ) expresses the dependence between and in its functional form and parameters. The Morgenstern family of distribution provides a general technique by which a bivariate distribution can be constructed directly using its marginal distributions and the correlation between the variates, (Morgenstern [5]). A generalization of Morgenstern method was proposed by Farlie [6] which is Balakrishnan and Johnson [19]. Aleem [14] introduced a bivariate inverted Weibull distribution using the Farlie-Gumbel-Morgenstern idea. In this paper, an attempt has been taken to introduce bivariate EIW distribution using this system of distribution. where ( ) and ( ) are the marginal pdf s of and .
Concomitant variables play a significant role to study order statistics for bivariate set up of distribution when the associated characteristic is not available or difficult to measure. It was first introduced by David [7] and also mentioned as the induced order statistic by Bhattacharya [17].
Concomitants of order statistics by using the concept of Morgenstern approach have been extensively used by several authors. Balasubramanian and Beg [11]

EXPONENTIATED INVERTED WEIBULL DISTRIBUTION (EIWD)
If has EIWD with shape parameters and and scale parameter then the distribution function is given by, The pdf is given by, The ℎ moments of the exponentiated inverted Weibull distribution is given as follows: Hence, the mean and variance of the EIWD is given by And,

Order Statistics of EIWD
Let us assume that 1 , 2 , … is a random sample from an absolutely continuous population with probability density function (pdf) ( ) and cumulative distribution function (cdf) ( ).
Let, 1: ≤ 2: ≤ ⋯ ≤ : be the order statistics obtained by arranging the preceding random sample in increasing order of magnitude, then the density function of : ; (1 ≤ ≤ ) be, Hence, using the cdf and pdf of EIWD as given by equations (3) and (4), we obtain the pdf of the ℎ order statistic : ; (1 ≤ ≤ ) as, putting = 1 and = in (9) we have the pdf s of the smallest and largest order statistics of EIWD as, The cdf of From equation (12) by using the cdf of EIWD as given by equation (3) we have the distribution function of : as By putting = 1 and = in equation (13) we get the distribution functions of the smallest and largest order statistics of EIWD as, The ℎ moment of We obtain the joint density function of two order statistics The joint cumulative distribution function of the order statistics : and : be, The ( , ) ℎ product moment of ( : , : ) of EIWD to be , :

BIVARIATE EXPONENTIATED INVERTED WEIBULL DISTRIBUTION (BEIWD)
In this section, we develop the bivariate exponentiated inverted Weibull distribution (BEIWD) using the Morgenstern approach. This system provides a very general expression of a bivariate distribution from which it can be derived by substituting any desired set of marginal distributions.
Since both the bivariate distribution function and density are given in terms of marginals, it is easy to generate a random sample from a Morgenstern distribution.
Let be the life time of a very expensive component of a two component system and be an inexpensive variable (directly measurable or observable) which is correlated with Y.
From equation (22), we can derive the regression curve of given = for MTBEIWD as Thus, it is seen that the conditional expectation is non-linear with respect to .
The product moment for the joint pdf where 1 ≤ < ≤ is given by Hence, the first order product moment between and is, So, the covariance term is given by, Therefore, the Karl-Pearson's co-efficient of correlation is obtained as, For Morgenstern family of distribution the Pearson's correlation coefficient lies between − 1 3 ⁄ and 1 3 ⁄ for desired values of parameters.
Survival function of Morgenstern family is of the form From equation (28) the survival function of BEIWD is given by, The plots of the survival function of BEIWD are displayed below. The plots indicate that the survival function decreases faster as the values of and increase. The conditional distribution function of given = for BEIWD is obtained as The conditional density function of given = for Morgenstern family is defined as, Now, by using equations (32) and (9) we get the cdf of the concomitant of ℎ order statistic : from equation (30) [ : ] ( ) = ∫ | ( | ) : ( ) Hence, putting = in equation (34) By putting = , in equation (37) By putting = 1, we get the pdf of the concomitant of smallest order statistic 1: which is,

Joint density of Concomitants of Two Order Statistics
To study the behavior of one concomitant to others we need the joint probability distribution of any two concomitants. In this section, we derive the joint distribution of concomitants of two From equation (40), we obtain,

Some properties of Concomitants of Order Statistics
In this section, the explicit expression for the moment of concomitant of ℎ order statistics is Using the result, we can compute mean and variance of the concomitant of ℎ order statistic.
According to Scaria and Nair [10], the density of the concomitant [ : ] can be also written as, Resulting from (43), the ℎ moment of [ : ] can be derived as, Hence, using equations (5) and (16), the ℎ moment of concomitant of order statistic [ : ] is obtained as,

SURVIVAL AND HAZARD FUNCTIONS OF CONCOMITANTS OF ORDER STATISTICS
When the random vector ( , ) is defined as the lifetimes of two units or lives at a certain time point with observable ages, the remaining lifetime probabilities can be computed by using the general definition of a survival function, ( ) = 1 − ( ) for each of ( , ).
The hazard function of concomitant of order statistic [ : ] is given by,   The behavior of the survival function [ : ] (y) of the concomitant [ : ] is presented in the graphics below.