par Ley, Christophe 
;Paindaveine, Davy ;Verdebout, Thomas
Référence Journal of Multivariate Analysis, 139, page (79-91)
Publication Publié, 2015
;Paindaveine, Davy ;Verdebout, Thomas Référence Journal of Multivariate Analysis, 139, page (79-91)
Publication Publié, 2015
Article révisé par les pairs
| Résumé : | This paper mainly focuses on one of the most classical testing problems in directional statistics, namely the spherical location problem that consists in testing the null hypothesis H0:θ=θ0 under which the (rotational) symmetry center θ is equal to a given value θ0. The most classical procedure for this problem is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its asymptotic theory requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more problems nowadays involve high-dimensional directional data (e.g., in genetics or text mining). In the present work, we derive the asymptotic null distribution of the Watson statistic as both n and p go to infinity. This reveals that (i) the Watson test is robust against high dimensionality, and that (ii) it allows for (n, p)-asymptotic results that are universal, in the sense that p may go to infinity arbitrarily fast (or slowly) as a function of n. Turning to Euclidean data, we show that our results also lead to a test for the null that the covariance matrix of a high-dimensional multinormal distribution has a "θ0-spiked" structure. Finally, Monte Carlo studies corroborate our asymptotic results and briefly explore non-null rejection frequencies. |
