## The Annals of Statistics

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

### Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems

Michael D. Perlman and Ingram Olkin

#### 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**

links.jstor.org

**Subjects**

Primary: 62H10: Distribution of statistics

Secondary: 62H15: Hypothesis testing 62H20: Measures of association (correlation, canonical correlation, etc.) 62J05: Linear regression 62J10: Analysis of variance and covariance

**Keywords**

Unbiasedness of invariant multivariate tests monotonicity of power functions noncentral Wishart matrix characteristic roots maximal invariants noncentral distributions hypergeometric function of matrix arguments stochastically increasing MANOVA growth curves model testing for independence canonical correlations complex multivariate normal distribution FKG inequality HPKE inequality positively associated random variables pairwise total positivity of order two rectangular coordinates

#### Citation

Perlman, Michael D.; Olkin, Ingram. Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems. Ann. Statist. 8 (1980), no. 6, 1326--1341. doi:10.1214/aos/1176345204. https://projecteuclid.org/euclid.aos/1176345204