CSAIL Event Calendar: Previous Series
Component Identification and Estimation in Nonlinear Multi-dimensional Regression Models by Structural Adaptation
Speaker: Alexander Samarov , Sloan School of Management, MIT, and Univ. of Massachusetts-Lowell
We propose a new method of analysis of partially linear models, whose nonlinear component is completely unknown but is assumed to have a small dimension. The target of analysis is identification of the set of predictors that enter in a nonlinear way in the model function, and the complete estimation of the model, including slope coefficients of the linear component and the link function of the nonlinear component. The method provides a procedure for selection of the significant variables in nonparametric regression. The method of analysis goes back to the idea of structural adaptation from Hristache et al., where the problem of dimension reduction was considered for a multiple-index model. Our approach is fully adaptive to the unknown model structure and requires very mild assumptions about the model. The only important assumption is that the dimensionality of nonlinear component is relatively small. Theoretical results indicate that our procedure identifies the nonlinear component with probability arbitrarily close to one as long as the nonlinear component is separated from linearity by the squared distance (variance) of the order log(n)/n or larger, where n is the sample size. After the predictors have been classified into linear and nonlinear, the slopes of the linear component and the nonparametric link function of the nonlinear component are estimated using the standard methods. This is joint work with V. Spokoiny and C. Vial.