Majorization bounds for distribution function

TL;DR

This paper develops bounds for a distribution function using majorization theory, expressing them as mixtures of order statistic distributions, and shows these bounds converge to the true distribution under certain conditions.

Contribution

It introduces a novel approach to bounding distribution functions via majorization, using mixtures of order statistic distributions, with proven convergence properties.

Findings

Bounds expressed as mixtures of order statistic distributions.

Convergence of bounds to the true distribution as parameters vary.

Application of majorization theory to distribution function estimation.

Abstract

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of distribution functions of order statistics, i.e. $\sum \limits_{i=1}^{n}p_{i}F_{i:n}$ and $\sum \limits_{i=1}^{n}p_{i}F_{n-i+1:n}.$ 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 $m\rightarrow\infty.$