M -type smoothing splines with auxiliary scale estimation

Abstract A robust data smoothing procedure is derived from coupling M -type smoothing splines with appropriate robust scale estimators. These robustified smoothing splines are computed using an iteratively reweighted penalized least-squares algorithm. A numerical example is given to illustrate the utility of the proposed methodology and demonstrate that the case weights deriving from the robustification process have important diagnostic uses. Asymptotic theory is also provided including consistency and rates of convergence for the proposed estimator.

[1]  T. Gasser,et al.  Residual variance and residual pattern in nonlinear regression , 1986 .

[2]  T. J. Carmody,et al.  Diagnostics for multivariate smoothing splines , 1988 .

[3]  R. L. Eubank,et al.  Diagnostics for penalized least-squares estimators , 1986 .

[4]  P. Speckman.  Spline Smoothing and Optimal Rates of Convergence in Nonparametric Regression Models , 1985 .

[5]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[6]  L. Schumaker,et al.  Computation of Smoothing and Interpolating Natural Splines via Local Bases , 1973 .

[7]  Arnold Zellner,et al.  Applied Time Series Analysis of Economic Data. , 1985 .

[8]  W. Härdle,et al.  Robust Non-parametric Function Fitting , 1984 .

[9]  M. Hutchinson,et al.  An efficient method for calculating smoothing splines using orthogonal transformations , 1986 .

[10]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[11]  R. L. Eubank,et al.  The hat matrix for smoothing splines , 1984 .

[12]  M. Hutchinson,et al.  Smoothing noisy data with spline functions , 1985 .

[13]  W. Härdle.  Robust regression function estimation , 1984 .

[14]  T. Gasser,et al.  On robust kernel estimation of derivatives of regression functions , 1985 .

[15]  R. L. Eubank,et al.  Diagnostics for Smoothing Splines , 1985 .

[16]  C. Reinsch.  Smoothing by spline functions , 1967 .

[17]  P. Laurent,et al.  A general method for the construction of interpolating or smoothing spline-functions , 1968 .

[18]  D. Cox.  Asymptotics for $M$-Type Smoothing Splines , 1983 .

[19]  B. Yandell,et al.  Automatic Smoothing of Regression Functions in Generalized Linear Models , 1986 .