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January, 1978 Bound on the Classification Error for Discriminating Between Multivariate Populations with Specified Means and Covariance Matrices
K. Isii, Y. Taga
Ann. Statist. 6(1): 132-141 (January, 1978). DOI: 10.1214/aos/1176344072

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.

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K. Isii. Y. Taga. "Bound on the Classification Error for Discriminating Between Multivariate Populations with Specified Means and Covariance Matrices." Ann. Statist. 6 (1) 132 - 141, January, 1978. https://doi.org/10.1214/aos/1176344072

Information

Published: January, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0377.62032
MathSciNet: MR468036
Digital Object Identifier: 10.1214/aos/1176344072

Subjects:
Primary: 62H30
Secondary: 62G99 , 90C05

Keywords: bound for error probability , classification rule , discrimination , minimax theorem

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 1 • January, 1978
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