Abstract: High-dimensional classification is an important and challenging statistical problem. We develop a set of quadratic discriminant rules by simplifying the structure of the covariance matrices instead of imposing sparsity assumptions — either on the covariance matrices themselves (or their inverses), or on the standardized between-class distance. Under moderate conditions on the population covariance matrices, our quadratic discriminant rules enjoy good asymptotic properties. Computationally, they are easy to implement and do not require large-scale mathematical programming. Numerically, they perform well in finite dimensions and with finite sample sizes. We present analyses of several classic micro-array data sets.

Key words and phrases: Asymptotic misclassification probability, classification, covariance matrix estimate, normality, unequal covariance matrices.