Bootstrap Testing of the Rank of a Matrix via Least-Squared Constrained Estimation
To test if an unknown matrix <italic>M</italic> <sub>0</sub> has a given rank (null hypothesis noted <italic>H</italic> <sub>0</sub>), we consider a statistic that is a squared distance between an estimator <inline-formula> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_847841_ilm0001.gif"/> </inline-formula> and the submanifold of fixed-rank matrix. Under <italic>H</italic> <sub>0</sub>, this statistic converges to a weighted chi-squared distribution. We introduce the constrained bootstrap (CS bootstrap) to estimate the law of this statistic under <italic>H</italic> <sub>0</sub>. An important point is that even if <italic>H</italic> <sub>0</sub> fails, the CS bootstrap reproduces the behavior of the statistic under <italic>H</italic> <sub>0</sub>. 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.
