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.
翻译:我们用投影追踪来研究线性差异的估计,这种方法是盲目的,因为它没有在估计中使用分类标签。我们的观点是空洞的,而作为我们的主要贡献,我们根据三种不同的预测指数(skewness, kurtsisis)及其组合来得出估计天花板的中央限值。结果显示,在每一种情况下,限制的共变矩阵与线性差异分析(LDA)(LDA)是成比例的,后者是一个非盲目的差异估计者。在对非随机差异进行的广泛比较研究显示,在群体被任意地分开,其大小合理平衡的情况下,预测追求能够达到与LDA相同的效率。我们以一个真正的数据实例和一个模拟研究来结束,调查为有限样品所获得的无干扰的公式的有效性。