Extremal Probabilistic Problems and Hotelling's T^2 Test Under Symmetry Condition

Abstract

We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T^2 tends to χ^2_d in distribution. We show that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ from the one based on χ^2_d and at the same time is applicable not only for large random homogeneous samples but for all multidimensional samples without exceptions. The main assertions have the form of inequalities, not that of limit theorems; these inequalities are exact representing the solutions to certain extremal problems. Let us also mention an auxiliary result which itself may be of interest: χ_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).

23 pages; a shorter version of this preprint appeared in Ann. Statist. Vol. 22 (1994) 357--368. Some papers refer to this preprint (rather than to the published version), especially to some unpublished parts of the preprint. (The article "a" is missing in the title of the preprint, but present in the title of the published version: "under a symmetry condition".)