  A number of settings arise in which it is of interest to predict Principal
Component (PC) scores for new observations using data from an initial sample.
In this paper, we demonstrate that naive approaches to PC score prediction can
be substantially biased toward 0 in the analysis of large matrices. This
phenomenon is largely related to known inconsistency results for sample
eigenvalues and eigenvectors as both dimensions of the matrix increase. For the
spiked eigenvalue model for random matrices, we expand the generality of these
results, and propose bias-adjusted PC score prediction. In addition, we compute
the asymptotic correlation coefficient between PC scores from sample and
population eigenvectors. Simulation and real data examples from the genetics
literature show the improved bias and numerical properties of our estimators.
