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.
Citation
C. L. Mallows. "Bounds on Distribution Functions in Terms of Expectations of Order- Statistics." Ann. Probab. 1 (2) 297 - 303, April, 1973. https://doi.org/10.1214/aop/1176996981
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