  Many sparse linear discriminant analysis (LDA) methods have been proposed to
overcome the major problems of the classic LDA in high-dimensional settings.
However, the asymptotic optimality results are limited to the case that there
are only two classes, which is due to the fact that the classification boundary
of LDA is a hyperplane and explicit formulas exist for the classification error
in this case. In the situation where there are more than two classes, the
classification boundary is usually complicated and no explicit formulas for the
classification errors exist. In this paper, we consider the asymptotic
optimality in the high-dimensional settings for a large family of linear
classification rules with arbitrary number of classes under the situation of
multivariate normal distribution. Our main theorem provides easy-to-check
criteria for the asymptotic optimality of a general classification rule in this
family as dimensionality and sample size both go to infinity and the number of
classes is arbitrary. We establish the corresponding convergence rates. The
general theory is applied to the classic LDA and the extensions of two recently
proposed sparse LDA methods to obtain the asymptotic optimality. We conduct
simulation studies on the extended methods in various settings.
