## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 3 (1975), 617-636.

### Exact Robustness Studies of Tests of Two Multivariate Hypotheses Based on Four Criteria and Their Distribution Problems Under Violations

#### Abstract

This paper deals with robustness studies of tests of two hypotheses (A) $\Sigma_1 = \Sigma_2$ in $N(\mu_i, \Sigma_i), i = 1, 2$, and (B) $\mu_1 = \cdots = \mu_l$ in $N(\mu_i, \Sigma), i = 1, 2, \cdots, l, \Sigma$ unknown, based on four test criteria (a) Hotelling's trace, (b) Pillai's trace, (c) Wilks' $\Lambda$ and (d) Roy's largest root. The robustness for (A) is against non-normality and for (B) against unequal covariance matrices and is studied in the exact case, unlike the results obtained earlier. In this connection, Pillai's density of the latent roots of $\mathbb{S}_1\mathbb{S}_2^{-1}$ under violations is used to derive the distributions or the moments of the criteria. Numerical studies of the tests of the two hypotheses based on the four criteria are made for the two-roots case.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 3 (1975), 617-636.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343126

**Digital Object Identifier**

doi:10.1214/aos/1176343126

**Mathematical Reviews number (MathSciNet)**

MR378250

**Zentralblatt MATH identifier**

0342.62032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H10: Distribution of statistics

Secondary: 62H15: Hypothesis testing

**Keywords**

Distribution problems under violations non-normality unequal covariance matrices Hotelling's trace Pillai's trace Wilks' criterion Roy's largest root exact robustness tests of multivariate hypotheses tabulations

#### Citation

Pillai, K. C. S.; Sudjana. Exact Robustness Studies of Tests of Two Multivariate Hypotheses Based on Four Criteria and Their Distribution Problems Under Violations. Ann. Statist. 3 (1975), no. 3, 617--636. doi:10.1214/aos/1176343126. https://projecteuclid.org/euclid.aos/1176343126