## The Annals of Statistics

- Ann. Statist.
- Volume 6, Number 1 (1978), 132-141.

### Bound on the Classification Error for Discriminating Between Multivariate Populations with Specified Means and Covariance Matrices

#### Abstract

Let $\mathscr{F}_1, \mathscr{F}_2$ be two families of $p$-variate distribution functions with specified means $\mathbf{\mu}_i (i = 1,2)$ and nonsingular covariance matrices $\Sigma_i$, and let $\pi_i$ be the prior probability assigned to $\mathscr{F}_i$ for $i = 1, 2$. The objective is to discriminate whether an observation $\mathbf{x}$ is from a distribution $F_1 \in \mathscr{F}_1$ or $F_2 \in \mathscr{F}_2$. Given a pair $F = (F_1, F_2)$ the error probability for classification rule $\phi$ is denoted by $e(\phi, F)$. In this paper the values of $\sup_F \inf_\phi e(\phi, F)$ and $\inf_\phi \sup_F e(\phi, F)$ are found and conditions for the existence of a saddle point of $e(\phi, F)$ are given. Also a saddle point is found when it exists. When $\phi$ is restricted to linear classification rules the same problems are considered. The mathematical programming method for finding a saddle point is also outlined.

#### Article information

**Source**

Ann. Statist., Volume 6, Number 1 (1978), 132-141.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344072

**Digital Object Identifier**

doi:10.1214/aos/1176344072

**Mathematical Reviews number (MathSciNet)**

MR468036

**Zentralblatt MATH identifier**

0377.62032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Secondary: 62G99: None of the above, but in this section 90C05: Linear programming

**Keywords**

Discrimination classification rule bound for error probability minimax theorem

#### Citation

Isii, K.; Taga, Y. Bound on the Classification Error for Discriminating Between Multivariate Populations with Specified Means and Covariance Matrices. Ann. Statist. 6 (1978), no. 1, 132--141. doi:10.1214/aos/1176344072. https://projecteuclid.org/euclid.aos/1176344072