  We study the statistical decision process of detecting the low-rank signal
from various signal-plus-noise type data matrices, known as the spiked random
matrix models. We first show that the principal component analysis can be
improved by entrywise pre-transforming the data matrix if the noise is
non-Gaussian, generalizing the known results for the spiked random matrix
models with rank-1 signals. As an intermediate step, we find out sharp phase
transition thresholds for the extreme eigenvalues of spiked random matrices,
which generalize the Baik-Ben Arous-P\'{e}ch\'{e} (BBP) transition. We also
prove the central limit theorem for the linear spectral statistics for the
spiked random matrices and propose a hypothesis test based on it, which does
not depend on the distribution of the signal or the noise. When the noise is
non-Gaussian noise, the test can be improved with an entrywise transformation
to the data matrix with additive noise. We also introduce an algorithm that
estimates the rank of the signal when it is not known a priori.
