# Non-equivariant Simultaneous Confidence Intervals Less Likely to Contain Zero

Report Number
411
Authors
Bin Yu
Citation
J. Am. Stat. Assoc., v91, pp329-337, 1996.
Abstract

We exhibit a procedure for finding simultaneous confidence intervals for the expectations $\fatmu=(\mu_j)_{j=1}^n$ of a set of independent random variables, identically distributed up to their location parameters, that yields intervals less likely to contain zero than the standard simultaneous confidence intervals for many $\fatmu \ne {\bf 0}$. The procedure is defined implicitly by inverting a non-equivariant hypothesis test with a hyperrectangular acceptance region whose orientation depends on the unsigned ranks of the components of $\fatmu$, then projecting the convex hull of the resulting confidence region onto the coordinate axes. The projection to obtain simultaneous confidence intervals implicitly involves solving $n!$ sets of linear inequalities in $n$ variables, but the optima are attained among a set of at most $n^2$ such sets, and can be found by a simple algorithm. The procedure also works when the statistics are exchangeable but not independent, and can be extended to cases where the inference is based on statistics for $\fatmu$ that are independent but not necessarily identically distributed, provided there are known functions of $\fatmu$ that are location parameters for the statistics. However, in the general case, it appears that all $n!$ sets of linear inequalities must be examined to find the confidence intervals.

PDF File
Postscript File