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In binary classification, imbalance refers to situations in which one classis heavily under-represented. This issue is due to either a data collectionprocess or because one class is indeed rare in a population. Imbalancedclassification frequently arises in applications such as biology, medicine,engineering, and social sciences. In this paper, for the first time, wetheoretically study the impact of imbalance class sizes on the lineardiscriminant analysis (LDA) in high dimensions. We show that due to datascarcity in one class, referred to as the minority class, andhigh-dimensionality of the feature space, the LDA ignores the minority classyielding a maximum misclassification rate. We then propose a new constructionof hard-thresholding rules based on a data splitting technique that reduces thelarge difference between the misclassification rates. We show that the proposedmethod is asymptotically optimal. We further study two well-known sparseversions of the LDA in imbalanced cases. We evaluate the finite-sampleperformance of different methods using simulations and by analyzing two realdata sets. The results show that our method either outperforms its competitorsor has comparable performance based on a much smaller subset of selectedfeatures, while being computationally more efficient.

New Hard-thresholding Rules based on Data Splitting in High-dimensional Imbalanced Classification