Asymptotic Distribution of the Largest Eigenvalue via Geometric Representations of High-Dimension, Low-Sample-Size Data

A common feature of high-dimensional data is that the data dimension is high, however, the sample size is relatively low. We call such a data HDLSS data. In this paper, we study HDLSS asymptotics for Gaussian-type HDLSS data. We find a surprising geometric representation of the HDLSS data in a dual space. We give an estimator of the eigenvalue by using the noise-reduction (NR) methodology. We show that the estimator enjoys consistency properties under mild conditions when the dimension is high. We provide an asymptotic distribution for the largest eigenvalue when the dimension is high while the sample size is fixed. We show that the estimator given by the NR methodology holds the asymptotic distribution under a condition milder than that for the conventional estimator.

DOI: http://dx.doi.org/10.4038/sljastats.v5i4.7785