Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation

TL;DR

This paper introduces a constrained bootstrap method for testing the rank of a matrix by estimating the distribution of a family of statistics based on squared distances, offering a practical and accurate alternative to traditional asymptotic tests.

Contribution

The paper develops a constrained bootstrap approach for rank testing that is consistent, easy to compute, and applicable to a broad class of sub-manifold hypotheses.

Findings

Constrained bootstrap accurately estimates the distribution under the null hypothesis.

The method outperforms traditional asymptotic approaches in simulations.

Applicable to general sub-manifold parameter testing.

Abstract

In order to test if an unknown matrix has a given rank (null hypothesis), we consider the family of statistics that are minimum squared distances between an estimator and the manifold of fixed-rank matrix. Under the null hypothesis, every statistic of this family converges to a weighted chi-squared distribution. In this paper, we introduce the constrained bootstrap to build bootstrap estimate of the law under the null hypothesis of such statistics. As a result, the constrained bootstrap is employed to estimate the quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light one the accuracy of the constrained bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test if an unknown parameter belongs to a sub-manifold locally smooth. Finally, the constrained bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.