## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 1 (1994), 357-368.

### Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition

#### Abstract

We consider the Hotelling $T^2$ statistic for an arbitrary $d$-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under the zero-means hypothesis the limiting distribution for $T^2$ is $\chi^2_d$. It is shown that a test for the orthant symmetry condition introduced by Efron can be constructed which does not differ essentially from the one based on $\chi^2_d$ and at the same time is applicable not only to large random homogeneous samples but to all multidimensional samples. The main results are not limit theorems, but exact inequalities corresponding to the solutions to certain extremal problems. The following auxiliary result itself may be of interest: $\chi_d - \sqrt{d - 1}$ has a monotone likelihood ratio.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 1 (1994), 357-368.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325373

**Digital Object Identifier**

doi:10.1214/aos/1176325373

**Mathematical Reviews number (MathSciNet)**

MR1272088

**Zentralblatt MATH identifier**

0812.62065

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H15: Hypothesis testing

Secondary: 60E15: Inequalities; stochastic orderings 62F04 62F35: Robustness and adaptive procedures 62G10: Hypothesis testing 62G15: Tolerance and confidence regions

**Keywords**

Hotelling's $T^2$ test probability inequalities extremal probability problems monotone likelihood ratio stochastic ordering

#### Citation

Pinelis, Iosif. Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition. Ann. Statist. 22 (1994), no. 1, 357--368. doi:10.1214/aos/1176325373. https://projecteuclid.org/euclid.aos/1176325373