Variable selection for high-dimensional varying coefficient partially linear models via nonconcave penalty

In this paper, we consider the problem of simultaneous variable selection and estimation for varying-coefficient partially linear models in a “small <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$n$$</EquationSource> </InlineEquation>, large <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation>” setting, when the number of coefficients in the linear part diverges with sample size while the number of varying coefficients is fixed. Similar problem has been considered in Lam and Fan (Ann Stat 36(5):2232–2260, <CitationRef CitationID="CR20">2008</CitationRef>) based on kernel estimates for the nonparametric part, in which no variable selection was investigated besides that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation> was assume to be smaller than <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$n$$</EquationSource> </InlineEquation>. Here we use polynomial spline to approximate the nonparametric coefficients which is more computationally expedient, demonstrate the convergence rates as well as asymptotic normality of the linear coefficients, and further present the oracle property of the SCAD-penalized estimator which works for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation> almost as large as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\exp \{n^{1/2}\}$$</EquationSource> </InlineEquation> under mild assumptions. Monte Carlo studies and real data analysis are presented to demonstrate the finite sample behavior of the proposed estimator. Our theoretical and empirical investigations are actually carried out for the generalized varying-coefficient partially linear models, including both Gaussian data and binary data as special cases. Copyright Springer-Verlag Berlin Heidelberg 2013