We consider high-dimensional quadratic classifiers in non-sparse settings. The target of classification rules is not Bayes error rates in the context. The classifier based on the Mahalanobis distance does not always give a preferable performance even if the populations are normal distributions having known covariance matrices. The quadratic classifiers proposed in this paper draw information about heterogeneity effectively through both the differences of expanding mean vectors and covariance matrices. We show that they hold a consistency property in which misclassification rates tend to zero as the dimension goes to infinity under non-sparse settings. We verify that they are asymptotically distributed as a normal distribution under certain conditions. We also propose a quadratic classifier after feature selection by using both the differences of mean vectors and covariance matrices. Finally, we discuss performances of the classifiers in actual data analyses. The proposed classifiers achieve highly accurate classification with very low computational costs.
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