Bivariate Tensor-Product B-Splines in a Partly Linear Model

In some applications, the mean or median response is linearly related to some variables but the relation to additional variables are not easily parameterized. Partly linear models arise naturally in such circumstances. Suppose that a random sample {(Ti, Xi, Yi),i=1, 2, ?, n} is modeled byYi=XTis0+g0(Ti)+errori, whereYiis a real-valued response,Xi?RpandTiranges over a unit square, andg0is an unknown function with a certain degree of smoothness. We make use of bivariate tensor-product B-splines as an approximation of the functiong0and consider M-type regression splines by minimization of ?ni=1?(Yi?XTis?gn(Ti)) for some convex function?. Mean, median and quantile regressions are included in this class. We show under appropriate conditions that the parameter estimate ofsachieves its information bound asymptotically and the function estimate ofg0attains the optimal rate of convergence in mean squared error. Our asymptotic results generalize directly to higher dimensions (for the variableT) provided that the functiong0is sufficiently smooth. Such smoothness conditions have often been assumed in the literature, but they impose practical limitations for the application of multivariate tensor product splines in function estimation. We also discuss the implementation of B-spline approximations based on commonly used knot selection criteria together with a simulation study of both mean and median regressions of partly linear models.

[1]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[2]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[3]  Stephen Portnoy,et al.  Bivariate quantile smoothing splines , 1998 .

[4]  Randall Eubank,et al.  Convergence rates for estimation in certain partially linear models , 1989 .

[5]  A. McQuarrie,et al.  Regression and Time Series Model Selection , 1998 .

[6]  J. Rice.  Convergence rates for partially splined models , 1986 .

[7]  P. Bickel,et al.  Achieving Information Bounds in Non and Semiparametric Models , 1990 .

[8]  C. J. Stone,et al.  Additive Regression and Other Nonparametric Models , 1985 .

[9]  R. Shiller,et al.  Smoothness Priors and Nonlinear Regression , 1982 .

[10]  Peide Shi,et al.  On the rates of convergence of minimum L1-norm' estimates in a partly linear model , 1994 .

[11]  Clifford M. Hurvich,et al.  Regression and time series model selection in small samples , 1989 .

[12]  Trevor Hastie,et al.  [Flexible Parsimonious Smoothing and Additive Modeling]: Discussion , 1989 .

[13]  J H Frieadman.  MULTIVARIATE ADDITIVE REGRESSION SPLINES , 1991 .

[14]  Xuming He,et al.  Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-Variables Models , 1997 .

[15]  Pin T. Ng,et al.  Quantile smoothing splines , 1994 .

[16]  Leo Breiman,et al.  [The ∏ Method for Estimating Multivariate Functions from Noisy Data]: Response , 1991 .

[17]  D. Varberg.  Convex Functions , 1973 .

[18]  Hung Chen,et al.  Polynomial splines and nonparametric regression , 1991 .

[19]  S. Portnoy.  Asymptotic behavior of M-estimators of p regression parameters when p , 1985 .

[20]  C. J. Stone,et al.  Logspline Density Estimation for Censored Data , 1992 .

[21]  P. Shi,et al.  Convergence rate of b-spline estimators of nonparametric conditional quantile functions ∗ , 1994 .

[22]  Hung Chen,et al.  A two-stage spline smoothing method for partially linear models , 1991 .

[23]  S. Portnoy.  Asymptotic Behavior of $M$-Estimators of $p$ Regression Parameters when $p^2/n$ is Large. I. Consistency , 1984 .

[24]  J. Shiau,et al.  Data-Driven Efficient Estimators for a Partially Linear Model , 1994 .

[25]  Xuming He.  Quantile Curves without Crossing , 1997 .

[26]  P. Speckman.  Kernel smoothing in partial linear models , 1988 .

[27]  W. J. Studden,et al.  Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines , 1980 .

[28]  G. Wahba,et al.  The computation of generalized cross-validation functions through householder tridiagonalization with applications to the fitting of interaction spline models , 1989 .

[29]  C. J. Stone,et al.  The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation , 1994 .

[30]  Jerome H. Friedman.  [The ∏ Method for Estimating Multivariate Functions from Noisy Data]: Discussion , 1991 .

[31]  Probal Chaudhuri,et al.  Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation , 1991 .

[32]  J. Friedman,et al.  FLEXIBLE PARSIMONIOUS SMOOTHING AND ADDITIVE MODELING , 1989 .

[33]  C. J. Stone,et al.  Asymptotics for Doubly Flexible Logspline Response Models , 1991 .

[34]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[35]  Nancy E. Heckman,et al.  Spline Smoothing in a Partly Linear Model , 1986 .

[36]  Frederick Mosteller,et al.  Data Analysis and Regression , 1978 .

[37]  A. A. Weiss,et al.  Semiparametric estimates of the relation between weather and electricity sales , 1986 .

[38]  R. Koenker,et al.  Hierarchical Spline Models for Conditional Quantiles and the Demand for Electricity , 1990 .

[39]  Zehua Chen.  Interaction Spline Models and Their Convergence Rates , 1991 .

[40]  Chong Gu.  [The ∏ Method for Estimating Multivariate Functions from Noisy Data]: Discussion , 1991 .

[41]  Hung Chen,et al.  Convergence Rates for Parametric Components in a Partly Linear Model , 1988 .

[42]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[43]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .