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.

翻译：本研究提出了一种创新方法，旨在改善Fisher线性判别分析（LDA）在高维数据分类场景中的性能，特别针对每个特征具有不同方差的情况。该方法利用非参数最大似然估计（NPMLE）技术来估计均值与方差参数。通过适应特征间方差的异质性，所提出的方法显著提升了分类性能。尤其值得注意的是，与众多先前研究假设异质方差服从右偏逆伽马分布不同，本方法在异质方差呈左偏、对称或右偏分布时均表现出卓越性能。我们开展了一系列严格的实验来实证验证该方法的有效性，结果表明所提方法能准确分类具有异质方差的高维数据。