BIVARIATE TRANSMUTED EXPONENTIATED GUMBEL DISTRIBUTION (BTEGD) AND CONCOMITANTS OF ITS ORDER STATISTICS

In this article we have studied bivariate transmuted exponentiated Gumbel distribution using Morgenstern approach (Morgenstern [4]). We have also studied the shape behavior of the pdf and cdf of the bivariate transmuted exponentiated Gumbel distribution. The distribution of the concomitants of rorder statistics, the moment generating function (mgf) and moments of the concomitants of r order statistics are obtained. Numerical computations have been done for the moments of the concomitants.


INTRODUCTION
The Gumbel distribution is a very popular statistical distribution due to its extensive applicability in several areas and its wide applications have been reported by Kotz and Nadarajah [16]. The 3564 DEEPSHIKHA DEKA, BHANITA DAS, UPAMA DEKA, BHUPEN KUMAR BARUAH applicability of Gumbel distribution in the area of climate modeling, for example: global warming problems, offshore modeling, rainfall and wind speed modeling have been discussed by Nadarajah [18]. In several areas of engineering such as: flood frequency analysis, network space, software reliability, structural and wind engineering, the applicability of Gumbel Distribution has been reported by Cardeiro, Ortega and Cunha [5]. Due to its wide applicability, several works aimed at extending the Gumbel distribution becomes important. Some examples are mentioned in: Nadarajah and Kotz [17], Cardeiro, Ortega and Cunha [5], Andrade, Rodrigues, Bourguignon and Cordeiro [22] and Deka, Das and Baruah [3]. Thus, the interest in theory and methods about the Gumbel distribution is progressive. Concomitants of order statistics have been used extensively by several authors using the concept of Morgenstern approach. Shahbaz and Shahbaz [19] have studied concomitants of generalized order statistics for a bivariate Weibull distribution. Tahmaseb and Jafari [21] have studied concomitants of order statistics and record values from Morgenstern type bivariate generalized exponential distribution. Khan and Kumar [14] have studied concomitants of order statistics from Weighted Marshall-Olkin Bivariate Exponential distribution. Chacko and Thomas [13] have studied estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling. Athar and Nayabuddin [10], have studied concomitants of dual generalized In the following sections, order statistics and concomitants are mentioned first and then the new Bivariate Transmuted Exponentiated Gumbel distribution is presented. A next step is the construction of the distribution of the concomitants of order statistics for the presented Bivariate Transmuted Exponentiated Gumbel distribution.
For a detailed overview of concomitants, we refer to David and Nagaraja [8], [9].
Concomitants of order statistics have several applications in statistics. Concomitants are used in many applied areas where a population characteristic is investigated with respect to another characteristic of the same population. For example in selection procedures, Yeo and David [23] considered the problem of choosing the best k objects out of n candidates on the basis of auxiliary Where : ( ) is the pdf of : .
The density function of an ℎ order statistic ( : ), is defined by Arnold et al. [2] as, The general expressions given here are used in the following sections where a new Bivariate Transmuted Exponentiated distribution is presented.

BIVARIATE TRANSMUTED EXPONENTIATED GUMBEL DISTRIBUTION
Deka et al. [3] have studied the Transmuted Exponentiated Gumbel Distribution (TEGD) along with several statistical properties and applied it to model water quality parameters data set. The cdf of the TEGD is And its corresponding pdf is Using the marginal Transmuted Exponentiated Gumbel density functions for the random variable and where ~( 1 , 1 , 1 , 1 ) and ~( 2 , 2 , 2 , 2 ) in equation (1) we get the cdf for MTBTEGD as And the corresponding pdf is obtained by using (2) as It can be shown that

Distribution of Concomitants for BTEGD
In this section, we obtain the distribution of the concomitants of the ℎ order statistics for the Bivariate Transmuted Exponentiated Gumbel distribution, given in Eq. (7). To obtain the distribution of the concomitants of ℎ order statistics we need the conditional distribution of given and the distribution of ℎ order statistics : . The conditional distribution of given is obtained as follows: We have, Using equation (6) and (8), in equation (9) we get the conditional distribution of given for BTEGD as Putting = 1, in the equation (3), we get the distribution of the concomitants of 1 st order statistics as Using series representation as Using Eq. (12) in Eq. (4), we get the expression for 1: ( ) as Using Prudnikov et al. [1] after integration equation (14) becomes 3572 DEEPSHIKHA DEKA, BHANITA DAS, UPAMA DEKA, BHUPEN KUMAR BARUAH According to David [7] the cdf of the order statistics connected by the relation Therefore the relation (*) is also true for pdf of order statistics also. Thus we can obtained the pdf of [ : ] from the following relation Now using equation (15) in equation (16) we get the pdf of the concomitants of ℎ order statistics as 3573 BIVARIATE TRANSMUTED EXPONENTIATED GUMBEL DISTRIBUTION

Moments of Concomitants of Order Statistics
In this section, we have deduced the expression for the moment of concomitants of ℎ order statistics when random variables ( , ); ( = 1,2,3, … , ) are i.i.d. and follows TEGD. Utilizing these results, we can compute means and variates of concomitants of ℎ order statistics.
According to David [6], the ℎ moment about origin of concomitants of ℎ order statistics [ : ] is given by Equation (23) is obtained by using Prudnikov, Brychkov and Marichev [1] Let Finally, by using (2.6.21.1) from Prudnikov et al. [1] in equation (24), and after integration we get Using (23) and (25) in (22), we get the expression for the ℎ moment of concomitants of ℎ order statistics as

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.