  Principal component analysis (PCA) is a widely used dimension reduction
method, but its performance is known to be non-robust to outliers. Recently,
product-PCA (PPCA) has been shown to possess the efficiency-loss free
ordering-robustness property: (i) in the absence of outliers, PPCA and PCA
share the same asymptotic distributions; (ii), in the presence of outliers,
PPCA is more ordering-robust than PCA in estimating the leading eigenspace.
PPCA is thus different from the conventional robust PCA methods, and may
deserve further investigations. In this article, we study the high-dimensional
statistical properties of the PPCA eigenvalues via the techniques of random
matrix theory. In particular, we derive the critical value for being distant
spiked eigenvalues, the limiting values of the sample spiked eigenvalues, and
the limiting spectral distribution of PPCA. Similar to the case of PCA, the
explicit forms of the asymptotic properties of PPCA become available under the
special case of the simple spiked model. These results enable us to more
clearly understand the superiorities of PPCA in comparison with PCA. Numerical
studies are conducted to verify our results.
