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