Concomitants and majorization bounds for bivariate distribution function

Abstract

Let ($X,Y)$ be a random vector with distribution function $F(x,y),$ and $(X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n)$ are independent copies of ($X,Y).$ Let $X_{i:n}$ be the $i$th order statistics constructed from the sample $X_1,X_2,...,X_n$ of the first coordinate of the bivariate sample and $Y_{[i:n]}$ be the concomitant of $X_{i:n}.$ Denote $F_{i:n}$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of joint distributions of order statistics and their concomitants, i.e. ${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{i:n}(x,y)$ and ${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{n-i+1:n}(x,y).$ It is shown that these bounds converge to $F$ for a particular sequence $(p_1(m),p_2(m),...,p_n(m)),m=1,2,..$ as $m\to \infty .$

arXiv comment7 pages

References in corpus (1)

• Majorization bounds for distribution function