TY - JOUR
T1 - An alternating determinationoptimization approach for an additive multi-index model
AU - Feng, Zhenghui
AU - ZHU, Lixing
N1 - Funding Information:
We thank Dr. Heng Peng for discussing this project, Prof. Lizhi Liao and Dr. Leihong Zhang for introducing some methods of optimization to us. The authors also thank the associate editor and three referees for their constructive comments and suggestions which led to a great improvement of an early manuscript in presentation. Lixing Zhu was supported by a grant from the Research Grants Council of Hong Kong , and an FRG grant from Hong Kong Baptist University , Hong Kong, China.
PY - 2012/6
Y1 - 2012/6
N2 - Sufficient dimension reduction techniques are to deal with curse of dimensionality when the underlying model is of a very general semiparametric multi-index structure and to estimate the central subspace spanned by the indices. However, the cost is that they can only identify the central subspace/central mean subspace and its dimension, rather than the indices themselves. In this paper, we investigate estimation for an additive multi-index model (AMM) that is of an additive structure with indices. The problem for AMM involves determining and estimating the nonparametric component functions and estimating the corresponding indices in the model. Different from the classical sufficient dimension reduction techniques in the estimation of the subspace and dimensionality determination, we propose a new penalized method to implement the estimation of component functions and of indices simultaneously. To this end, we suggest an alternating determinationoptimization algorithm to alternatively fit best model and estimate the indices. Estimation consistency is provided. Simulation studies are carried out to examine the performance of the new method and a real data example is also analysed for illustration.
AB - Sufficient dimension reduction techniques are to deal with curse of dimensionality when the underlying model is of a very general semiparametric multi-index structure and to estimate the central subspace spanned by the indices. However, the cost is that they can only identify the central subspace/central mean subspace and its dimension, rather than the indices themselves. In this paper, we investigate estimation for an additive multi-index model (AMM) that is of an additive structure with indices. The problem for AMM involves determining and estimating the nonparametric component functions and estimating the corresponding indices in the model. Different from the classical sufficient dimension reduction techniques in the estimation of the subspace and dimensionality determination, we propose a new penalized method to implement the estimation of component functions and of indices simultaneously. To this end, we suggest an alternating determinationoptimization algorithm to alternatively fit best model and estimate the indices. Estimation consistency is provided. Simulation studies are carried out to examine the performance of the new method and a real data example is also analysed for illustration.
KW - Bayesian information criterion
KW - Dimension reduction
KW - Hierarchical type LASSO
KW - Projected gradient method
KW - Spline approximation
UR - http://www.scopus.com/inward/record.url?scp=84857655775&partnerID=8YFLogxK
U2 - 10.1016/j.csda.2011.12.004
DO - 10.1016/j.csda.2011.12.004
M3 - Journal article
AN - SCOPUS:84857655775
SN - 0167-9473
VL - 56
SP - 1981
EP - 1993
JO - Computational Statistics and Data Analysis
JF - Computational Statistics and Data Analysis
IS - 6
ER -