  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 ${\pmb \theta}_1^0$ 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 hypothesis, hence
also under sequences of contiguous alternatives. We show that this equivalence
does not survive asymptotic scenarios where
$\lambda_{n1}/\lambda_{n2}=1+O(r_n)$ 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.
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 $r_n$ to converge to zero at an arbitrary rate. While
we restrict to single-spiked spectra of the form
$\lambda_{n1}>\lambda_{n2}=\ldots=\lambda_{np}$ 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.
