In this study, we introduce an innovative methodology aimed at enhancing Fisher's Linear Discriminant Analysis (LDA) in the context of high-dimensional data classification scenarios, specifically addressing situations where each feature exhibits distinct variances. Our approach leverages Nonparametric Maximum Likelihood Estimation (NPMLE) techniques to estimate both the mean and variance parameters. By accommodating varying variances among features, our proposed method leads to notable improvements in classification performance. In particular, unlike numerous prior studies that assume the distribution of heterogeneous variances follows a right-skewed inverse gamma distribution, our proposed method demonstrates excellent performance even when the distribution of heterogeneous variances takes on left-skewed, symmetric, or right-skewed forms. We conducted a series of rigorous experiments to empirically validate the effectiveness of our approach. The results of these experiments demonstrate that our proposed methodology excels in accurately classifying high-dimensional data characterized by heterogeneous variances.

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