基于投影寻踪的线性判别盲估计的大样本性质
Large-Sample Properties of Blind Estimation of the Linear Discriminant Using Projection Pursuit
AI总结：
本文从渐近角度研究不使用类别标签的投影寻踪线性判别盲估计，针对偏度、峰度及其凸组合三种投影指标推导中心极限定理，证明其极限协方差矩阵与LDA成比例，并揭示在组间充分分离且规模平衡时投影寻踪可达到与LDA相同的效率。
摘要：
我们研究使用投影寻踪方法对线性判别进行估计，该方法是一种盲方法，即在估计过程中不使用类别标签。我们的视角是渐近的，作为主要贡献，我们为基于三种不同投影指标的估计量推导了中心极限定理，这三种指标分别是偏度、峰度及其凸组合。结果表明，在每种情况下，极限协方差矩阵都与线性判别分析（LDA）的极限协方差矩阵成比例，而LDA是判别的一种非盲估计量。对渐近方差的广泛比较研究揭示，当各组任意良好分离且各组规模合理平衡时，投影寻踪能够达到与LDA相等的效率。最后，我们以一个真实数据示例和一项模拟研究作为结论，该模拟研究考察了所获得的渐近公式在有限样本下的有效性。
英文摘要：
We study the estimation of the linear discriminant with projection pursuit, a method that is blind in the sense that it does not use the class labels in the estimation. Our viewpoint is asymptotic and, as our main contribution, we derive central limit theorems for estimators based on three different projection indices, skewness, kurtosis and their convex combination. The results show that in each case the limiting covariance matrix is proportional to that of linear discriminant analysis (LDA), an unblind estimator of the discriminant. An extensive comparative study between the asymptotic variances reveals that projection pursuit is able to achieve efficiency equal to LDA when the groups are arbitrarily well-separated and their sizes are reasonably balanced. We conclude with a real data example and a simulation study investigating the validity of the obtained asymptotic formulas for finite samples.
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