## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 4 (1974), 694-702.

### Admissibility of Translation Invariant Tolerance Intervals in the Location Parameter Case

#### Abstract

Given $n$ independent observations with common density $f(x - \theta)$, and a rv $z$ independent of these with density $g(x - \theta) (f, g$ known except for $\theta$) a prediction region for $z$ is required. It is shown that the best translation invariant interval is optimal in two senses: (1) there is no other region with the same expected coverage (coverage is the probability of containing $z$) and uniformly smaller expected size (Lebesgue measure); (2) no other interval having the same confidence that the coverage exceeds $\beta$ (given) can have uniformly smaller expected length. The best invariant interval in each case is found, and the normal case is studied. The usual interval centered at $\bar{X}$ is not always optimal in the second sense if $\beta$ and/or confidence are small. A criterion involving expected coverage and the confidence of exceeding coverage $\beta$ is also examined. Again restrictions on these are needed for the usual normal interval to be optimal.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 4 (1974), 694-702.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342757

**Digital Object Identifier**

doi:10.1214/aos/1176342757

**Mathematical Reviews number (MathSciNet)**

MR370900

**Zentralblatt MATH identifier**

0297.62024

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F25: Tolerance and confidence regions

Secondary: 62C15: Admissibility 62C10: Bayesian problems; characterization of Bayes procedures

**Keywords**

Tolerance intervals admissibility normal tolerance intervals prediction regions

#### Citation

Blumenthal, Saul. Admissibility of Translation Invariant Tolerance Intervals in the Location Parameter Case. Ann. Statist. 2 (1974), no. 4, 694--702. doi:10.1214/aos/1176342757. https://projecteuclid.org/euclid.aos/1176342757