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.

In many applications, especially in cancer treatment and diagnosis, investigators are interested in classifying patients into various diagnosis groups on the basis of molecular data such as gene expression or proteomic data. Often, some of the diagnosis groups are known to be related to higher or lower values of some of the predictors. The standard methods of classifying patients into various groups do not take into account the underlying order. This could potentially result in high misclassification rates, especially when the number of groups is larger than two. In this article, we develop classification procedures that exploit the underlying order among the mean values of the predictor variables and the diagnostic groups by using ideas from order-restricted inference. We generalize the existing methodology on discrimination under restrictions and provide empirical evidence to demonstrate that the proposed methodology improves over the existing unrestricted methodology. The proposed methodology is applied to a bladder cancer data set where the researchers are interested in classifying patients into various groups.

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