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Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems

Michael D. Perlman and Ingram Olkin
The Annals of Statistics
Vol. 8, No. 6 (Nov., 1980), pp. 1326-1341
Stable URL: http://www.jstor.org/stable/2240945
Page Count: 16
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Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems
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Abstract

Let Y:p × r and Z:p × n be normally distributed random matrices whose r + n columns are mutually independent with common covariance matrix, and EZ = 0. It is desired to test μ = 0 vs. μ ≠ 0, where μ = EY. Let d1, ⋯, dp denote the characteristic roots of YY'(YY' + ZZ')-1. It is shown that any test with monotone acceptance region in d1, ⋯, dp, i.e., a region of the form {g(d1, ⋯, dp)≤ c} where g is nondecreasing in each argument, is unbiased. Similar results hold for the problems of testing independence of two sets of variates, for the generalized MANOVA (growth curves) model, and for analogous problems involving the complex multivariate normal distribution. A partial monotonicity property of the power functions of such tests is also given.

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