Structure discovery and parametrically guided regression

Abstract

In regression analysis, a parametric model is often assumed from prior information or a pilot study. If the model assumption is valid, the parametric method is useful. However, the efficiency of the estimator is not guaranteed when a poor model is selected. This article aims to check whether the model assumption is correct or not and to estimate the regression function. To achieve this, we propose a hybrid technique of parametrically guided method and group lasso. First, the parametric model is prepared. The parametrically guided estimator is constructed by summing the parametric estimator and nonparametric estimator. For the estimation of the nonparametric component, we use B‐splines and the group lasso method. If the nonparametric component is estimated to be a zero function, the parametrically guided estimator is reduced to the parametric estimator. Then, we can decide that the parametric model assumption is correct. If the nonparametric estimator is not zero, the semiparametric estimator is obtained. Thus, the proposed method discovers the model structure and estimates the regression function simultaneously. We investigate the asymptotic properties of the proposed estimator. A simulation study and real data example are presented. Copyright © 2016 John Wiley & Sons, Ltd.