New efficient spline estimation for varying-coefficient models with two-step knot number selection
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New efficient spline estimation for varying-coefficient models with two-step knot number selection Jun Jin1 · Tiefeng Ma1 · Jiajia Dai2 Received: 9 December 2019 / Accepted: 22 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract One of the advantages for the varying-coefficient model is to allow the coefficients to vary as smooth functions of other variables and the coefficients functions can be estimated easily through a simple B-spline approximations method. This leads to a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when some coefficient functions possess different degrees of smoothness. Under the regularity conditions, the consistency and asymptotic normality of the two step B-spline estimators are also derived. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an AIDS data set. Keywords Varying-coefficient models · Asymptotic normality · B-spline · Adaptive knot selection · Two-step B-spline
1 Introduction Varying-coefficient models (VCM), as a generalization of linear models, have widely been used to explore the complicated relationship between a response and predictors of interest. The VCM, proposed by Hastie and Tibshirani (1993), has the form Y = X T β(U ) + ε,
(1.1)
where β(u) = (β1 (u), . . . , β p (u))T is a p × 1 vector of unknown functions. X , U are covariates , Y is the response variable, and ε is the model error with E(ε|X , U ) = 0. In addition, to avoid the curse of dimensionality, it is usually assumed that U is a scalar.
B
Jun Jin [email protected]
1
Center of Statistical Research, School of Statistics, Southwestern University of Finance and Economics, Chengdu 661130, China
2
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China
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J. Jin et al.
Without loss of generality, it is assumed to be the unit interval [0, 1]. An advantage of model (1.1) is that by allowing the coefficients β(u) to depend on U , the modeling bias can be reduced significantly. Over the past decades, much effort has been devoted to studying the methodological, theoretical and applied sides around varying-coefficient models. A two-step estimation procedure was proposed by Fan and Zhang (1999) to deal with the situations where coefficient functions admit different degrees of smoothness. Fan and Zhang (2000) studied the confidence band construction and hypothesis testing of the coefficient function. Cai et al. (2000) studied the estimation and verification of generalized variable coefficient models based on local polynomials. Zhang and Lee (2000) considered the variable bandwidth selection problem of local polynomial estimation for the coefficient function. Chiang et al. (2001) proposed a smoothing spline estimation approach for the coefficient functions. Huang et al. (2002) developed a global smoothing procedure using basis function approximations to estimate β(u). Eubank et al. (2004) used the smooth spline method t
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