An Effective Semiparametric Estimation Approach for the Sufficient Dimension Reduction Model

ABSTRACT

In the exploratory data analysis, the sufficient dimension reduction model has been widely used to characterize the conditional distribution of interest. Different from the existing approaches, our main achievement is to simultaneously estimate two essential elements, basis and structural dimension, of the central subspace and the bandwidth of a kernel distribution estimator through a single estimation criterion. With an appropriate order of kernel function, the proposed estimation procedure can be effectively carried out by starting with a dimension of zero until the first local minimum is reached. Meanwhile, the optimal bandwidth selector is ensured to be a valid tuning parameter for the central subspace estimator. An important advantage of this estimation technique is its flexibility to allow a response to be discrete and some of covariates to be discrete or categorical providing that a certain continuity condition holds. Under very mild assumptions, we further derive the uniform consistency of the introduced optimization function and the consistency of the resulting estimators. Moreover, the asymptotic normality of the central subspace estimator is established with an estimated rather than exact structural dimension. In extensive simulations, the developed approach generally outperforms the competitors. Data from previous studies are also used to illustrate the proposal. On the whole, our methodology is very effective in estimating the central subspace and conditional distribution, highly flexible in adapting diverse types of a response and covariates, and practically feasible in obtaining an asymptotically optimal and valid bandwidth estimator. Supplementary materials for this article are available online.

Keywords:

• Asymptotic normality
• Central subspace
• Consistency
• Cross-validation estimation
• Inverse regression estimation
• Optimal bandwidth
• Pseudo-least integrated squares estimation
• Semiparametric efficiency bound
• Semiparametric estimation
• Structural dimension
• Sufficient dimension reduction

Supplementary Materials

The online supplement contains the appendices for the article.

Acknowledgments

The research of the corresponding author was partially supported by the Ministry of Science and Technology grants 99-2118-M-002-003 and 100-2118-M-002-005-MY2 (Taiwan). The authors are grateful to Dr. Shao-Hsuan Wang for some numerical computations used in this article. We would also like to thank the associate editor and three reviewers for many constructive comments on this article .