Asymptotic and bootstrap tests for subspace dimension

TL;DR

This paper develops asymptotic and bootstrap statistical tests for determining the dimension of signal subspaces in linear dimension reduction methods like PCA, FOBI, and SIR, with theoretical and practical validation.

Contribution

It introduces novel bootstrap strategies and consistent test-based estimators for subspace dimension, enhancing small sample analysis in dimension reduction.

Findings

Bootstrap tests outperform asymptotic tests in small samples

Consistent estimators accurately identify subspace dimension

Simulations and real data validate proposed methods

Abstract

Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are compared in simulations and illustrated in real data examples.