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

Sparse estimation via nonconcave penalized likelihood in a factor analysis model