Certain bivariate distributions and random processes connected with maxima and minima
Keywords:Copula, distribution theory, exponentiated distribution, extremes, generalized exponential distribution, order statistics, random minimum, random maximum, Sibuya distribution
It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the
distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where alpha> 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions.