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

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

2007-01-28

arxiv.org/abs/math/0701806v1

Mathematics

Statistics Theory

23 pages; a shorter version of this preprint appeared in Ann. Statist. Vol. 22 (1994) 357--368. Some papers refer to this prep

Scientific paper

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 \chi^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 \chi^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: \chi_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).

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