Confidence Bounds for a set of Means

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.