The Annals of Statistics

Admissibility of Translation Invariant Tolerance Intervals in the Location Parameter Case

Saul Blumenthal

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F25: Tolerance and confidence regions
Secondary: 62C15: Admissibility 62C10: Bayesian problems; characterization of Bayes procedures

Tolerance intervals admissibility normal tolerance intervals prediction regions


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.

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