## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 30, Number 4 (1959), 926-938.

### Optimum Tolerance Regions and Power When Sampling From Some Non-Normal Universes

#### Abstract

We assume familiarity with the concepts defined in [1] and [2], where optimum $\beta$-expectation tolerance regions and their power functions were found for $k$-variate normal distributions. The method used is to reduce this problem to that of solving an equivalent hypothesis testing problem. It is the purpose of this paper to find optimum $\beta$-expectation tolerance regions for the single and double exponential distributions, and to exhibit the corresponding power functions. Let $X = (X_1, \cdots, X_n)$ be a random sample point in $n$ dimensions, where each $X_i$ is an independent observation, distributed by some continuous probability distribution function. It is often desirable to estimate on the basis of such a sample point a region, say $S(X_1, \cdots, X_n)$, which contains a given fraction $\beta$ of the parent distribution. We usually seek to estimate the center 100 $\beta$% of the distribution and/or one of the 100 $\beta$% tails of the parent distribution.

#### Article information

**Source**

Ann. Math. Statist., Volume 30, Number 4 (1959), 926-938.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706076

**Digital Object Identifier**

doi:10.1214/aoms/1177706076

**Mathematical Reviews number (MathSciNet)**

MR109399

**Zentralblatt MATH identifier**

0089.15303

**JSTOR**

links.jstor.org

#### Citation

Guttman, Irwin. Optimum Tolerance Regions and Power When Sampling From Some Non-Normal Universes. Ann. Math. Statist. 30 (1959), no. 4, 926--938. doi:10.1214/aoms/1177706076. https://projecteuclid.org/euclid.aoms/1177706076