  This article studies the robustness of the eigenvalue ordering, an important
issue when estimating the leading eigen-subspace by principal component
analysis (PCA). In Yata and Aoshima (2010), cross-data-matrix PCA (CDM-PCA) was
proposed and shown to have smaller bias than PCA in estimating eigenvalues.
While CDM-PCA has the potential to achieve better estimation of the leading
eigen-subspace than the usual PCA, its robustness is not well recognized. In
this article, we first develop a more stable variant of CDM-PCA, which we call
product-PCA (PPCA), that provides a more convenient formulation for theoretical
investigation. Secondly, we prove that, in the presence of outliers, PPCA is
more robust than PCA in maintaining the correct ordering of leading
eigenvalues. The robustness gain in PPCA comes from the random data partition,
and it does not rely on a data down-weighting scheme as most robust statistical
methods do. This enables us to establish the surprising finding that, when
there are no outliers, PPCA and PCA share the same asymptotic distribution.
That is, the robustness gain of PPCA in estimating the leading eigen-subspace
has no efficiency loss in comparison with PCA. Simulation studies and a face
data example are presented to show the merits of PPCA. In conclusion, PPCA has
a good potential to replace the role of the usual PCA in real applications
whether outliers are present or not.
