On Minimum Variance Among Certain Linear Functions of Order Statistics

Abstract

Suppose there are $n$ normal populations $N(\mu_i, 1), i = 1, \cdots, n$ and that one random observation from each of these $n$ populations is given. Let $x_1 \leqq x_2 \leqq \cdots \leqq x_n$ be the observations when arranged in order of magnitude and let the corresponding $n$ random variables be denoted by $X_i, i = 1, \cdots, n.$ The following theorem is proved: THEOREM. \begin{equation*}\operatorname{Var}\big(\sum^n_{i = 1} c_i X_i\big), \text{where}\end{equation*}\begin{equation*}\tag{1}\sum^n_{i = 1} c_i = 1,\end{equation*} is minimum when $c_i = 1/n, i = 1, \cdots, n.$ The above theorem may be applied to provide a direct proof of the result that $\sum^n_{i = 1}X_i$ is the best unbiased linear function of order statistics for estimating the sum $\sum^n_{i = 1}\mu_i.$