Incorporating Additional Information to Normal Linear Discriminant Rules

Abstract

The most useful and broadly known rule in the classical two-group linear normal discriminant analysis is Anderson's rule. In this article we propose some alternative procedures that prove useful when prior constraints on the mean vectors are known. These rules are based on new estimators of the difference of means. We prove under mild conditions that the new rules perform better when the common covariance matrix is known. Simulated experiments show that the misclassification errors are lower for the restricted rules defined here in the general case of an unknown covariance matrix. The prior constraints on the mean vector restrict the parameter space to a cone. A family of estimators indexed by a parameter γ, with 0 ≤ γ ≤ 1, is defined using an iterative procedure in such a way that the estimator with a higher value for γ takes values closer to the center of the cone with a greater probability. When γ = 0, the restricted maximum likelihood estimator is given, although the most interesting rule from a theoretical and practical standpoint is obtained when the estimator chosen is given by γ = 1. The usefulness of the proposed rules with real data is demonstrated by their application to two medical examples, the first dealing with heart attack patients and the second dealing with a diabetes dataset. In the former case, restrictions among surviving and nonsurviving patients are used; in the latter, the restrictions arise from differences between the healthy and diabetic populations.