par Paindaveine, Davy 
;Remy, Julien ;Verdebout, Thomas
Référence Annals of statistics, 48, 1, page (324-345)
Publication Publié, 2020-12-01
;Remy, Julien ;Verdebout, Thomas Référence Annals of statistics, 48, 1, page (324-345)
Publication Publié, 2020-12-01
Article révisé par les pairs
| Résumé : | We consider the problem of testing, on the basis of a p-variate Gaussian random sample, the null hypothesis H0 : θ1 = θ0 1 against the alternative H1 : θ1 /= θ0 1, where θ1 is the “first” eigenvector of the underlying covariance matrix and θ0 1 is a fixed unit p-vector. In the classical setup where eigenvalues λ1 > λ2 ≥ · · · ≥ λp are fixed, the Anderson (Ann. Math. Stat. 34 (1963) 122–148) likelihood ratio test (LRT) and the Hallin, Paindaveine and Verdebout (Ann. Statist. 38 (2010) 3245–3299) Le Cam optimal test for this problem are asymptotically equivalent under the null hypothesis, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where λn 1/λn2 = 1 + O(rn) with rn = O(1/√n). For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT severely overrejects the null hypothesis. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam's asymptotic theory of statistical experiments, we study the non-null and optimality properties of the Le Cam optimal test in the aforementioned asymptotic scenarios and show that the null robustness of this test is not obtained at the expense of power. Our asymptotic investigation is extensive in the sense that it allows rn to converge to zero at an arbitrary rate. While we restrict to single-spiked spectra of the form λn 1 > λn2 = · · · = λnp to make our results as striking as possible, we extend our results to the more general elliptical case. Finally, we present an illustrative real data example. |
