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07 February 2025
 
  » arxiv » 1109.0141

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Majorization bounds for distribution function
Ismihan Bairamov ;
Date 1 Sep 2011
AbstractLet $X$ 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 $m ightarrowinfty.$
Source arXiv, 1109.0141
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