Robust Estimation for Generalized Additive Models

This article studies M-type estimators for fitting robust generalized additive models in the presence of anomalous data. A new theoretical construct is developed to connect the costly M-type estimation with least-squares type calculations. Its asymptotic properties are studied and used to motivate a computational algorithm. The main idea is to decompose the overall M-type estimation problem into a sequence of well-studied conventional additive model fittings. The resulting algorithm is fast and stable, can be paired with different nonparametric smoothers, and can also be applied to cases with multiple covariates. As another contribution of this article, automatic methods for smoothing parameter selection are proposed. These methods are designed to be resistant to outliers. The empirical performance of the proposed methodology is illustrated via both simulation experiments and real data analysis. Supplementary materials are available online.

[1]  Christophe Croux,et al.  Robust Estimation of Mean and Dispersion Functions in Extended Generalized Additive Models , 2010, Biometrics.

[2]  D. Ruppert,et al.  Optimally bounded score functions for generalized linear models with applications to logistic regression , 1986 .

[3]  Douglas W. Nychka,et al.  Splines as Local Smoothers , 1995 .

[4]  R. Carroll,et al.  Segmented regression with errors in predictors: semi-parametric and parametric methods. , 1997, Statistics in medicine.

[5]  R. Tibshirani,et al.  Generalized Additive Models , 1986 .

[6]  Alan Y. Chiang,et al.  Generalized Additive Models: An Introduction With R , 2007, Technometrics.

[7]  M. Stone.  Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[8]  R. Carroll,et al.  On Robustness in the Logistic Regression Model , 1993 .

[9]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[10]  G. Kitagawa,et al.  Generalised information criteria in model selection , 1996 .

[11]  B. Ripley,et al.  Semiparametric Regression: Preface , 2003 .

[12]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[13]  J S Preisser,et al.  Robust Regression for Clustered Data with Application to Binary Responses , 1999, Biometrics.

[14]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[15]  R. Bhansali,et al.  Some properties of the order of an autoregressive model selected by a generalization of Akaike∘s EPF criterion , 1977 .

[16]  S. Morgenthaler.  Least-Absolute-Deviations Fits for Generalized Linear Models , 1992 .

[17]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[18]  Elvezio Ronchetti,et al.  Resistant selection of the smoothing parameter for smoothing splines , 2001, Stat. Comput..

[19]  S. Wood,et al.  Generalized Additive Models: An Introduction with R , 2006 .

[20]  Douglas W. Nychka,et al.  The Role of Pseudo Data for Robust Smoothing with Application to Wavelet Regression , 2007 .

[21]  I. Gijbels,et al.  Robust Estimation of Mean and Dispersion Functions in Extended Generalized Additive Models , 2010, Biometrics.

[22]  Christine M. Anderson-Cook,et al.  Generalized Additive Models: An Introduction With R , 2007 .

[23]  G. Wahba.  Spline models for observational data , 1990 .

[24]  S. Wood.  Generalized Additive Models: An Introduction with R , 2006 .

[25]  J. Copas.  Binary Regression Models for Contaminated Data , 1988 .

[26]  P. J. Huber.  Robust Regression: Asymptotics, Conjectures and Monte Carlo , 1973 .

[27]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[28]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[29]  C. Jennison,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[30]  Göran Kauermann,et al.  Generalized Cross-Validation for Bandwidth Selection of Backfitting Estimates in Generalized Additive Models , 2004 .

[31]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[32]  S. Wood.  Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models , 2004 .

[33]  E. Ronchetti,et al.  Robust Inference for Generalized Linear Models , 2001 .

[34]  R. Carroll,et al.  Conditionally Unbiased Bounded-Influence Estimation in General Regression Models, with Applications to Generalized Linear Models , 1989 .

[35]  Matias Salibian-Barrera,et al.  An Outlier-Robust Fit for Generalized Additive Models With Applications to Disease Outbreak Detection , 2011 .