## The Annals of Mathematical Statistics

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

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

#### 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