  Quadratic discriminant analysis (QDA) is a simple method to classify a
subject into two populations, and was proven to perform as well as the Bayes
rule when the data dimension p is fixed. The main purpose of this paper is to
examine the empirical and theoretical behaviors of QDA where p grows
proportionally to the sample sizes without imposing any structural assumption
on the parameters. The first finding in this moderate dimension regime is that
QDA can perform as poorly as random guessing even when the two populations
deviate significantly. This motivates a generalized version of QDA that
automatically adapts to dimensionality. Under a finite fourth moment condition,
we derive misclassification rates for both the generalized QDA and the optimal
one. A direct comparison reveals one "easy" case where the difference between
two rates converges to zero and one "hard" case where that converges to some
strictly positive constant. For the latter, a divide-and-conquer approach over
dimension (rather than sample) followed by a screening procedure is proposed to
narrow the gap. Various numerical studies are conducted to back up the proposed
methodology.
