## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 8, Number 6 (1980), 1326-1341.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345204

Digital Object Identifier
doi:10.1214/aos/1176345204

Mathematical Reviews number (MathSciNet)
MR594648

Zentralblatt MATH identifier
0465.62046

JSTOR