Bootstrap testing of the rank of a matrix via least-squared constrained estimation

• Autores: François Portier, Bernard Delyon
• Localización: Journal of the American Statistical Association, ISSN 0162-1459, Vol. 109, Nº 505, 2014, págs. 160-172
• Idioma: inglés
• DOI: 10.1080/01621459.2013.847841
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• Resumen

  • 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.