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July, 1974 Admissibility of Translation Invariant Tolerance Intervals in the Location Parameter Case
Saul Blumenthal
Ann. Statist. 2(4): 694-702 (July, 1974). DOI: 10.1214/aos/1176342757


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.


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Saul Blumenthal. "Admissibility of Translation Invariant Tolerance Intervals in the Location Parameter Case." Ann. Statist. 2 (4) 694 - 702, July, 1974.


Published: July, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0297.62024
MathSciNet: MR370900
Digital Object Identifier: 10.1214/aos/1176342757

Primary: 62F25
Secondary: 62C10 , 62C15

Keywords: Admissibility , normal tolerance intervals , prediction regions , tolerance intervals

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 4 • July, 1974
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