December, 1952 Confidence Bounds for a set of Means
D. A. S. Fraser
Ann. Math. Statist. 23(4): 575-585 (December, 1952). DOI: 10.1214/aoms/1177729336

## Abstract

Professor John Tukey suggested the following two problems to the author: given that $X_1, X_2, \cdots, X_n$ are normally and independently distributed with unknown means $\mu_1, \mu_2, \cdots, \mu_n$ and given variance $\sigma^2$; PROBLEM A: Find a $\beta$-level confidence interval of the form $g(x_1, \cdots, x_n) \geqq \mu_1, \cdots, \mu_n \geqq - \infty.$ PROBLEM B: Find a $\beta$-level confidence interval of the form $g(x_1, \cdots, x_n) \geqq \mu_1, \cdots, \mu_n \geqq h(x_1, \cdots, x_n).$ The main result of the paper is the nonexistence of intervals satisfying mild regularity conditions and having an exact confidence level (unless $n = 1$ or $\beta = 0, 1)$. However for each problem an interval is given for which the confidence level is greater than or equal to $\beta$ (formulas (2.1), (4.1)); these intervals are apparently shorter than those previously used in practice. Also the procedure for obtaining any interval with at least $\beta$ confidence is described. Some results are discussed for distributions other than the normal.

## Citation

D. A. S. Fraser. "Confidence Bounds for a set of Means." Ann. Math. Statist. 23 (4) 575 - 585, December, 1952. https://doi.org/10.1214/aoms/1177729336

## Information

Published: December, 1952
First available in Project Euclid: 28 April 2007

zbMATH: 0047.38003
MathSciNet: MR53440
Digital Object Identifier: 10.1214/aoms/1177729336 