  In many social, economical, biological and medical studies, one objective is
to classify a subject into one of several classes based on a set of variables
observed from the subject. Because the probability distribution of the
variables is usually unknown, the rule of classification is constructed using a
training sample. The well-known linear discriminant analysis (LDA) works well
for the situation where the number of variables used for classification is much
smaller than the training sample size. Because of the advance in technologies,
modern statistical studies often face classification problems with the number
of variables much larger than the sample size, and the LDA may perform poorly.
We explore when and why the LDA has poor performance and propose a sparse LDA
that is asymptotically optimal under some sparsity conditions on the unknown
parameters. For illustration of application, we discuss an example of
classifying human cancer into two classes of leukemia based on a set of 7,129
genes and a training sample of size 72. A simulation is also conducted to check
the performance of the proposed method.
