The Annals of Mathematical Statistics

On Minimum Variance Among Certain Linear Functions of Order Statistics

K. C. Seal

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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 $\sum^n_{i = 1}\mu_i.$

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Ann. Math. Statist., Volume 27, Number 3 (1956), 854-855.

First available in Project Euclid: 28 April 2007

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Seal, K. C. On Minimum Variance Among Certain Linear Functions of Order Statistics. Ann. Math. Statist. 27 (1956), no. 3, 854--855. doi:10.1214/aoms/1177728196.

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