Majorization bounds for distribution function
Abstract
Let X be a random variable with distribution function F, and X1,X2,...,Xn are independent copies of X. Consider the order statistics Xi:n, i=1,2,...,n and denote Fi:n(x)=P\Xi:n≤ 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. Σ i=1npiFi:n and Σ i=1npiFn-i+1:n. It is shown that these bounds converge to F \ for a particular sequence (p1(m),p2(m),...,pn(m)),m=1,2,.. as m→∞.
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