  We consider the problem of sparse estimation in a factor analysis model. A
traditional estimation procedure in use is the following two-step approach: the
model is estimated by maximum likelihood method and then a rotation technique
is utilized to find sparse factor loadings. However, the maximum likelihood
estimates cannot be obtained when the number of variables is much larger than
the number of observations. Furthermore, even if the maximum likelihood
estimates are available, the rotation technique does not often produce a
sufficiently sparse solution. In order to handle these problems, this paper
introduces a penalized likelihood procedure that imposes a nonconvex penalty on
the factor loadings. We show that the penalized likelihood procedure can be
viewed as a generalization of the traditional two-step approach, and the
proposed methodology can produce sparser solutions than the rotation technique.
A new algorithm via the EM algorithm along with coordinate descent is
introduced to compute the entire solution path, which permits the application
to a wide variety of convex and nonconvex penalties. Monte Carlo simulations
are conducted to investigate the performance of our modeling strategy. A real
data example is also given to illustrate our procedure.
