Robust variable selection in partially varying coefficient single-index model

By combining basis function approximations and smoothly clipped absolute deviation (SCAD) penalty, this paper proposes a robust variable selection procedure for a partially varying coefficient single-index model based on modal regression. The proposed procedure simultaneously selects significant variables in the parametric components and the nonparametric components. With appropriate selection of the tuning parameters, we establish the theoretical properties of our procedure, including consistency in variable selection and the oracle property in estimation. Furthermore, we also discuss the bandwidth selection and propose a modified expectation-maximization (EM)-type algorithm for the proposed estimation procedure. The finite sample properties of the proposed estimators are illustrated by some simulation examples.The research of Zhu is partially supported by National Natural Science Foundation of China (NNSFC) under Grants 71171075, 71221001 and 71031004. The research of Yu is supported by NNSFC under Grant 11261048

Robust variable selection for nonlinear models with diverging number of parameters

We focus on the problem of simultaneous variable selection and estimation for nonlinear models based on modal regression (MR), when the number of coefficients diverges with sample size. With appropriate selection of the tuning parameters, the resulting estimator is shown to be consistent and to enjoy the oracle properties

Nonparametric and Varying Coefficient Modal Regression

In this article, we propose a new nonparametric data analysis tool, which we call nonparametric modal regression, to investigate the relationship among interested variables based on estimating the mode of the conditional density of a response variable Y given predictors X. The nonparametric modal regression is distinguished from the conventional nonparametric regression in that, instead of the conditional average or median, it uses the "most likely" conditional values to measures the center. Better prediction performance and robustness are two important characteristics of nonparametric modal regression compared to traditional nonparametric mean regression and nonparametric median regression. We propose to use local polynomial regression to estimate the nonparametric modal regression. The asymptotic properties of the resulting estimator are investigated. To broaden the applicability of the nonparametric modal regression to high dimensional data or functional/longitudinal data, we further develop a nonparametric varying coefficient modal regression. A Monte Carlo simulation study and an analysis of health care expenditure data demonstrate some superior performance of the proposed nonparametric modal regression model to the traditional nonparametric mean regression and nonparametric median regression in terms of the prediction performance.Comment: 33 page

Modelling beyond Regression Functions: an Application of Multimodal Regression to Speed-Flow Data

An enormous amount of publications deals with smoothing in the sense of nonparametric regression. However, nearly all of the literature treats the case where predictors and response are related in the form of a function y=m(x)+noise. In many situations this simple functional model does not capture adequately the essential relation between predictor and response. We show by means of speed-flow diagrams, that a more general setting may be required, allowing for multifunctions instead of only functions. It turns out that in this case the conditional modes are more appropriate for the estimation of the underlying relation than the commonly used mean or the median. Estimation is achieved using a conditional mean-shift procedure, which is adapted to the present situation