Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems

Abstract

Let $Y:p \times r$ and $Z:p \times n$ be normally distributed random matrices whose $r + n$ columns are mutually independent with common covariance matrix, and $EZ = 0$. It is desired to test $\mu = 0$ vs. $\mu \neq 0$, where $\mu = EY$. Let $d_1, \cdots, d_p$ denote the characteristic roots of $YY'(YY' + ZZ')^{-1}$. It is shown that any test with monotone acceptance region in $d_1, \cdots, d_p$, i.e., a region of the form $\{g(d_1, \cdots, d_p)\leq c\}$ 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.