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Testing for Principal Component Directions under Weak Identifiability. (arXiv:1710.05291v2 [math.ST] UPDATED)
来源于:arXiv
We consider the problem of testing, on the basis of a $p$-variate Gaussian
random sample, the null hypothesis ${\cal H}_0:{\pmb \theta}_1= {\pmb
\theta}_1^0$ against the alternative ${\cal H}_1: {\pmb \theta}_1 \neq {\pmb
\theta}_1^0$, where ${\pmb \theta}_1$ is the "first" eigenvector of the
underlying covariance matrix and $\thetab_1^0$ is a fixed unit $p$-vector. In
the classical setup where eigenvalues $\lambda_1>\lambda_2\geq \ldots\geq
\lambda_p$ are fixed, the Anderson (1963) likelihood ratio test (LRT) and the
Hallin, Paindaveine, and Verdebout (2010) Le Cam optimal test for this problem
are asymptotically equivalent under the null, hence also under sequences of
contiguous alternatives. We show that this equivalence does not survive
asymptotic scenarios where $\lambda_{n1}-\lambda_{n2}=o(r_n)$ with
$r_n=O(1/\sqrt{n})$. For such scenarios, the Le Cam optimal test still
asymptotically meets the nominal level constraint, whereas the LRT severely
overrejects the null hypothesis. 查看全文>>