  Classification using high-dimensional features arises frequently in many
contemporary statistical studies such as tumor classification using microarray
or other high-throughput data. The impact of dimensionality on classifications
is poorly understood. In a seminal paper, Bickel and Levina [Bernoulli 10
(2004) 989--1010] show that the Fisher discriminant performs poorly due to
diverging spectra and they propose to use the independence rule to overcome the
problem. We first demonstrate that even for the independence classification
rule, classification using all the features can be as poor as the random
guessing due to noise accumulation in estimating population centroids in
high-dimensional feature space. In fact, we demonstrate further that almost all
linear discriminants can perform as poorly as the random guessing. Thus, it is
important to select a subset of important features for high-dimensional
classification, resulting in Features Annealed Independence Rules (FAIR). The
conditions under which all the important features can be selected by the
two-sample $t$-statistic are established. The choice of the optimal number of
features, or equivalently, the threshold value of the test statistics are
proposed based on an upper bound of the classification error. Simulation
studies and real data analysis support our theoretical results and demonstrate
convincingly the advantage of our new classification procedure.
