Model Selection and Estimation in Additive Regression Models

We propose a method of simultaneous model selection and estimation in additive regression models (ARMs) forindependent normal data. We use the mixed model representation of the smoothing spline estimators of thenonparametric functions in ARMs, where the importance of these functions is controlled by treating theinverse of the smoothing parameters as extra variance components. The selection of important nonparametricfunctions is achieved by maximizing the penalized likelihood with an adaptive LASSO. A unified EM algorithmis provided to obtain the maximum penalized likelihood estimates of the nonparametric functions and theresidual variance. In the same framework, we also consider forward selection based on score tests, and a twostage approach that imposes an early stage screening using an individual score test on each induced variancecomponent of the smoothing parameter.For longitudinal data, we propose to extend the adaptive LASSO and the two-stage selection with score testscreening to the additive mixed models (AMMs), by introducing subject-specific random effects to the additivemodels to accommodate the correlation in responses. We use the eigenvalue-eigenvector decomposition approachto approximate the working random effects in the linear mixed model presentation of the AMMs, so as to reducethe dimensions of matrices involved in the algorithm while keeping most data information, hence to tackle thecomputational problems caused by large sample sizes in longitudinal data.Simulation studies are provided and the methods are illustrated with data applications.