Concomitants and majorization bounds for bivariate distribution function

TL;DR

This paper develops bounds for the bivariate distribution function using majorization theory, involving mixtures of order statistics and their concomitants, and shows these bounds converge to the true distribution under certain conditions.

Contribution

It introduces a novel approach to bounding bivariate distribution functions via majorization, using mixtures of order statistics and concomitants, with proven convergence properties.

Findings

Bounds expressed as mixtures of joint distributions of order statistics and concomitants.

Bounds converge to the true distribution function as the mixture weights vary.

Provides a theoretical framework for approximation of bivariate distributions.

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 $m\rightarrow\infty.$