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 $F_{i:n}% (x,y)=P\{X_{i:n}\leq x,Y_{[i:n]}\leq y\}.$ 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 $m\rightarrow\infty.$