  In binary classification, imbalance refers to situations in which one class
is heavily under-represented. This issue is due to either a data collection
process or because one class is indeed rare in a population. Imbalanced
classification frequently arises in applications such as biology, medicine,
engineering, and social sciences. In this paper, for the first time, we
theoretically study the impact of imbalance class sizes on the linear
discriminant analysis (LDA) in high dimensions. We show that due to data
scarcity in one class, referred to as the minority class, and
high-dimensionality of the feature space, the LDA ignores the minority class
yielding a maximum misclassification rate. We then propose a new construction
of hard-thresholding rules based on a data splitting technique that reduces the
large difference between the misclassification rates. We show that the proposed
method is asymptotically optimal. We further study two well-known sparse
versions of the LDA in imbalanced cases. We evaluate the finite-sample
performance of different methods using simulations and by analyzing two real
data sets. The results show that our method either outperforms its competitors
or has comparable performance based on a much smaller subset of selected
features, while being computationally more efficient.
