Automatic Smoothing of Regression Functions in Generalized Linear Models

Abstract We consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models (Nelder and Wedderburn 1972) and present a generalized cross-validation procedure for empirically assessing an appropriate amount of smoothing in these estimates. Asymptotic arguments and numerical simulations are used to show that the generalized cross-validatory procedure preforms well from the point of view of a weighted mean squared error criterion. The methodology adds to the battery of graphical tools for model building and checking within the generalized linear model framework. Included are two examples motivated by medical and horticultural applications.

[1]  W. Rudin.  Principles of mathematical analysis , 1964 .

[2]  I. Good.  Nonparametric roughness penalties for probability densities , 1971 .

[3]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[4]  G. Wahba,et al.  A completely automatic french curve: fitting spline functions by cross validation , 1975 .

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

[6]  B. Silverman,et al.  Density Ratios, Empirical Likelihood and Cot Death , 1978 .

[7]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[8]  Tom Leonard.  Density Estimation, Stochastic Processes and Prior Information , 1978 .

[9]  J. Meinguet.  Multivariate interpolation at arbitrary points made simple , 1979 .

[10]  G. Wahba,et al.  Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation , 1980 .

[11]  V. Klema.  LINPACK user's guide , 1980 .

[12]  W. Raynor,et al.  High blood pressure and 17-year cancer mortality in the Western Electric Health Study. , 1981, American journal of epidemiology.

[13]  G. Wahba.  Spline Interpolation and Smoothing on the Sphere , 1981 .

[14]  J. Wendelberger,et al.  The Computation of Laplacian Smoothing Splines with Examples. , 1981 .

[15]  Jerome H. Friedman,et al.  Smoothing of Scatterplots , 1982 .

[16]  J. Anderson,et al.  Penalized maximum likelihood estimation in logistic regression and discrimination , 1982 .

[17]  Tom Leonard.  An Empirical Bayesian Approach to the Smooth Estimation of Unknown Functions. , 1982 .

[18]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[19]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[20]  D. M. Titterington,et al.  Cross-validation in nonparametric estimation of probabilities and probability densities , 1984 .

[21]  Peter J. Bickel,et al.  S: An Interactive Environment for Data Analysis and Graphics , 1984 .

[22]  A. Bowman.  An alternative method of cross-validation for the smoothing of density estimates , 1984 .

[23]  Ker-Chau Li,et al.  From Stein's Unbiased Risk Estimates to the Method of Generalized Cross Validation , 1985 .

[24]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.