Some Bounds for Expected Values of Order Statistics

Abstract

Let the function $F(x)$ be a distribution function for a continuous symmetric distribution, and let $X_{(i)}$ represent the $i$th order statistics from a sample of size $n$. It is shown in this paper that for $i \geqq (n + 1)/2$ $E(X_{(i)}) \geqq G(i/(n + 1)) \quad\text{if} F \text{is unimodal}$ and $E(X_{(i)}) \leqq G(i/(n + 1)) \quad\text{if} F is U\text{shaped},$ where $x = G(u)$ is the inverse function of $F(x) = u$. The definitions of unimodal and $U$-shaped distributions are given in Section 3. The above inequalities are of interest, since it is known (Blom (1958), Chapters 5 and 6) that for sufficiently large $n$ the bound $G(i/(n + 1))$ approaches $E(X_{(i)})$.