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

Abstract

To test if an unknown matrix M₀ has a given rank (null hypothesis noted H₀), we consider a statistic that is a squared distance between an estimator and the submanifold of fixed-rank matrix. Under H₀, this statistic converges to a weighted chi-squared distribution. We introduce the constrained bootstrap (CS bootstrap) to estimate the law of this statistic under H₀. An important point is that even if H₀ fails, the CS bootstrap reproduces the behavior of the statistic under H₀. As a consequence, the CS bootstrap is employed to estimate the nonasymptotic quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light on the accuracy of the CS bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test whether an unknown parameter belongs to a submanifold of the Euclidean space. Finally, the CS bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.

KEY WORDS:

• Dimension reduction
• Hypothesis testing
• Rank estimation