弱主方向的符号检验
Sign tests for weak principal directions
摘要：
我们考虑$p$ -变量椭圆分布的第一主方向上的推断。我们这样做，在具有挑战性的双渐近的情况下，这个方向最终无法识别。为了实现鲁棒性，不仅对于这种弱的可识别性，而且对于重尾，我们专注于基于符号的统计程序，也就是说，在程序中，涉及的意见，只有通过他们的方向从中心的分布。我们实际上考虑了检验原假设的一般问题，即第一主方向与$\mathbb{R}^p$的给定方向一致。我们首先集中在弱可识别性设置涉及单尖峰（即，涉及光谱的最小特征值具有多重性$p-1$）。我们表明，无论弱可识别性的程度，这样的设置提供了本地的替代品，相应的序列的统计实验收敛在Le Cam意义上。有趣的是，极限实验依赖于弱可识别性的程度。我们利用这个收敛结果，建立最佳的符号检验的问题考虑。在经典的渐近的情况下，频谱是固定的，这些测试被证明是渐近等价的符号为基础的似然比检验在文献中。然而，与后者不同的是，所提出的符号检验对任意弱的可识别性是鲁棒的。我们表明，我们的测试满足渐近水平的约束，无论频谱的结构，因此也可能在多尖峰设置。在弱可辨识性下，我们充分刻画了相应的检验统计量的非零渐近分布，这使得我们能够量化相应的局部渐近幂。
英文摘要：
We consider inference on the first principal direction of a $p$-variate elliptical distribution. We do so in challenging double asymptotic scenarios for which this direction eventually fails to be identifiable. In order to achieve robustness not only with respect to such weak identifiability but also with respect to heavy tails, we focus on sign-based statistical procedures, that is, on procedures that involve the observations only through their direction from the center of the distribution. We actually consider the generic problem of testing the null hypothesis that the first principal direction coincides with a given direction of $\mathbb{R}^p$. We first focus on weak identifiability setups involving single spikes (that is, involving spectra for which the smallest eigenvalue has multiplicity $p-1$). We show that, irrespective of the degree of weak identifiability, such setups offer local alternatives for which the corresponding sequence of statistical experiments converges in the Le Cam sense. Interestingly, the limiting experiments depend on the degree of weak identifiability. We exploit this convergence result to build optimal sign tests for the problem considered. In classical asymptotic scenarios where the spectrum is fixed, these tests are shown to be asymptotically equivalent to the sign-based likelihood ratio tests available in the literature. Unlike the latter, however, the proposed sign tests are robust to arbitrarily weak identifiability. We show that our tests meet the asymptotic level constraint irrespective of the structure of the spectrum, hence also in possibly multi-spike setups. We fully characterize the non-null asymptotic distributions of the corresponding test statistics under weak identifiability, which allows us to quantify the corresponding local asymptotic powers.
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