On the Expectation of Errors of Allocation Associated with a Linear Discriminant Function

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Exact expressions are obtained for the expectations of the actual, apparent and estimated errors of allocation which result when a linear discriminant function is used to allocate individuals to one of two multivariate normal populations with a known common dispersion matrix. Approximations suggested in the literature for the expectation of the actual error of allocation are compared with the exact values.

On Kolmogorov asymptotics of estimators of the misclassification error rate in linear discriminant analysis

“…Moran (1975) showed that (by adapting the results stated there to the definition of discriminant used here) E( ε̂ p ) = P ( Y < 0), where Y is a random variable distributed as (1 + ρ ) Z 1 − (1 − ρ ) Z 2 in which $Z_1~{\chi }_p^{2}false({\nu }_{1}false)$ and $Z_2~{\chi }_p^{2}false({\nu }_{2}false)$ are independent and…”

“…Here, following Raudys (1967), John (1961), and Moran (1975), we assume that the covariance matrix, Σ , is known and fixed; in particular, the W statistic is not a function of the sample covariance matrix, Σ̂ . In practice, however, if Σ is not known, then Σ̂ may be plugged in as an estimator of Σ .…”

“…We denote it by ε̂ p . After simplification, we have ε̂ p = Φ (− δ̂ /2) as given by Moran (1975), where $\hat{\delta }=\sqrt{{false({\overline{X}}_0-{\overline{X}}_{1}false)}^T{\mathrm{\sum }}^{-1}false({\overline{X}}_0-{\overline{X}}_{1}false)}$ is the estimated Mahalanobis distance between the classes.…”

“…Using simulation studies, Moran (1975) and Lachenbruch and Mickey (1968) stated that this estimator behaves similarly to resubstitution. We confirm this claim analytically here.…”

“…The successful applicability of a designed classifier relies on the measures of its predictive power, namely the actual or true error (Hills, 1966; Moran, 1975). However, in practice it is almost always the case that exactly evaluating the true error of a designed classifier is virtually impossible due to the lack of knowledge about the underlying distribution of the data.…”