### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 863-875 |

Number of pages | 13 |

Journal | Statistics and Computing |

Volume | 25 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 28 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Coordinate descent algorithm
- Factor analysis
- Nonconvex penalty
- Penalized likelihood
- Rotation technique

### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Computational Theory and Mathematics

### Cite this

**Sparse estimation via nonconcave penalized likelihood in factor analysis model.** / Hirose, Kei; Yamamoto, Michio.

Research output: Contribution to journal › Article

*Statistics and Computing*, vol. 25, no. 5, pp. 863-875. https://doi.org/10.1007/s11222-014-9458-0

}

TY - JOUR

T1 - Sparse estimation via nonconcave penalized likelihood in factor analysis model

AU - Hirose, Kei

AU - Yamamoto, Michio

PY - 2014/5/28

Y1 - 2014/5/28

N2 - 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.

AB - 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.

KW - Coordinate descent algorithm

KW - Factor analysis

KW - Nonconvex penalty

KW - Penalized likelihood

KW - Rotation technique

UR - http://www.scopus.com/inward/record.url?scp=84938416978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938416978&partnerID=8YFLogxK

U2 - 10.1007/s11222-014-9458-0

DO - 10.1007/s11222-014-9458-0

M3 - Article

AN - SCOPUS:84938416978

VL - 25

SP - 863

EP - 875

JO - Statistics and Computing

JF - Statistics and Computing

SN - 0960-3174

IS - 5

ER -