Moments of Order Statistics
次序统计量的矩
Let $X_{(i\mid n)}$ be the $i$th smallest order statistic from a population with $\operatorname{pdf} f(x)$ and $\operatorname{cdf} F(x)$. When $\tilde{x}$ is the population median, $n$ is the sample size and $G(x) = F(x)(1 - F(x))$, the following are proved. \begin{equation*}\begin{split}E(X_{(n-i+1\mid n)}) + E(X_{(i\mid n)}) \\ = n(n - 2i + 1)\binom{n - 1}{i - 1} \sum^{\lbrack (n-2i+1)/2\rbrack}_{\nu = 0}\frac{(-1)^\nu}{n - 2i + 1 - \nu}\binom{n - 2i + 1 - \nu}{\nu} \\ \times \Psi_1(i - 1 + \nu),\end{split}\end{equation*}…