Bounds on Distribution Functions in Terms of Expectations of Order- Statistics

Abstract

Suppose $x_1, \cdots, x_n$ are the order-statistics of a random sample from a distribution $F$. We assume that the expectations $\xi_{i:n} = E(x_i)$ are known, and derive sharp bounds on $F(x)$ for all $x$. These results are obtained by transforming the problem into a classical one involving ordinary power moments.