### Abstract

In factor regression model, the maximum likelihood estimation suffers from three disadvantages: (i) the maximum likelihood estimates are unavailable when the number of variables exceeds the number of observations, (ii) the rotation technique based on maximum likelihood estimates produces an insufficiently sparse loading matrix, and (iii) multicollinearity can occur when the estimates of unique variances (specific variances) are small because the regression coefficients are sensitive to the inverse of unique variances. To handle these problems, we propose a penalized maximum likelihood procedure. Specifically, we impose a lasso-type penalty on the factor loadings to improve the sparseness of the solution. We also introduce a penalty on unique variances, which (given the factor scores) corresponds to the ridge penalty on the regression coefficient. Theoretical properties from a prediction viewpoint of our procedure are discussed. The effectiveness of the procedure is investigated through Monte Carlo simulations. The utility of our procedure is demonstrated on real data collected by an online questionnaire.

Original language | English |
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Pages (from-to) | 633-662 |

Number of pages | 30 |

Journal | Statistical Papers |

Volume | 59 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Statistical Papers*,

*59*(2), 633-662. https://doi.org/10.1007/s00362-016-0781-8

**Sparse factor regression via penalized maximum likelihood estimation.** / Hirose, Kei; Imada, Miyuki.

Research output: Contribution to journal › Article

*Statistical Papers*, vol. 59, no. 2, pp. 633-662. https://doi.org/10.1007/s00362-016-0781-8

}

TY - JOUR

T1 - Sparse factor regression via penalized maximum likelihood estimation

AU - Hirose, Kei

AU - Imada, Miyuki

PY - 2018/6/1

Y1 - 2018/6/1

N2 - In factor regression model, the maximum likelihood estimation suffers from three disadvantages: (i) the maximum likelihood estimates are unavailable when the number of variables exceeds the number of observations, (ii) the rotation technique based on maximum likelihood estimates produces an insufficiently sparse loading matrix, and (iii) multicollinearity can occur when the estimates of unique variances (specific variances) are small because the regression coefficients are sensitive to the inverse of unique variances. To handle these problems, we propose a penalized maximum likelihood procedure. Specifically, we impose a lasso-type penalty on the factor loadings to improve the sparseness of the solution. We also introduce a penalty on unique variances, which (given the factor scores) corresponds to the ridge penalty on the regression coefficient. Theoretical properties from a prediction viewpoint of our procedure are discussed. The effectiveness of the procedure is investigated through Monte Carlo simulations. The utility of our procedure is demonstrated on real data collected by an online questionnaire.

AB - In factor regression model, the maximum likelihood estimation suffers from three disadvantages: (i) the maximum likelihood estimates are unavailable when the number of variables exceeds the number of observations, (ii) the rotation technique based on maximum likelihood estimates produces an insufficiently sparse loading matrix, and (iii) multicollinearity can occur when the estimates of unique variances (specific variances) are small because the regression coefficients are sensitive to the inverse of unique variances. To handle these problems, we propose a penalized maximum likelihood procedure. Specifically, we impose a lasso-type penalty on the factor loadings to improve the sparseness of the solution. We also introduce a penalty on unique variances, which (given the factor scores) corresponds to the ridge penalty on the regression coefficient. Theoretical properties from a prediction viewpoint of our procedure are discussed. The effectiveness of the procedure is investigated through Monte Carlo simulations. The utility of our procedure is demonstrated on real data collected by an online questionnaire.

UR - http://www.scopus.com/inward/record.url?scp=84969895684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969895684&partnerID=8YFLogxK

U2 - 10.1007/s00362-016-0781-8

DO - 10.1007/s00362-016-0781-8

M3 - Article

AN - SCOPUS:84969895684

VL - 59

SP - 633

EP - 662

JO - Statistical Papers

JF - Statistical Papers

SN - 0932-5026

IS - 2

ER -