The Annals of Mathematical Statistics

On Minimum Variance Among Certain Linear Functions of Order Statistics

K. C. Seal

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

Article information

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

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728196

Digital Object Identifier
doi:10.1214/aoms/1177728196

Mathematical Reviews number (MathSciNet)
MR79870

Zentralblatt MATH identifier
0074.13702

JSTOR
links.jstor.org

Citation

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. https://projecteuclid.org/euclid.aoms/1177728196


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