On Concomitants of Some Ordered Random Variables and Their Applications in Bivariate Modelling Problems

Concomitants of some ordered random variables

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

In this paper all existing results in the available literature on concomitants of order statistics, concomitants of lower and upper record values and concomitants of lower and upper generalized record values which characterize some bivariate distributions are reviewed. Applications of these results in modelling the parent bivariate distribution based on data sets available from it are further illustrated. A theory by which the distribution of the concomitant of the largest order statistic on one variable and the marginal distribution on the other variable are impounded together to generate the parent bivariate distribution and the theory by which impounding the distribution of the concomitant of the smallest order statistic with the marginal distribution on the other variable to discover the parent bivariate distribution as well are reviewed. A review of the extension of the above stated results by impounding the distribution of the concomitant of the upper generalized record value with the marginal distribution on the other variable to determine the parent bivariate distribution and by impounding the distribution of the concomitant of lower generalized record value with the marginal distribution on the other variable as well to determine the parent bivariate distributions are considered. Illustration of the determination of the parent bivariate distribution to each of the cases described above by specific examples as well are elucidated.