Abstract

This paper presents a fully Bayesian approach to regression splines with automatic knot selection in generalized semiparametric models for fundamentally non--Gaussian responses. In a basis function representation of the regression spline we use a B--spline basis. The reversible jump Markov chain Monte Carlo method allows for simultaneous estimation both of the number of knots and the knot placement, together with the unknown basis coefficients determining the shape of the spline. Since the spline can be represented as design matrix times unknown (basis) coefficients, it is straightforward to include additionally a vector of covariates with fixed effects, yielding a semiparametric model. The method is illustrated with data sets from the literature for curve estimation in generalized linear models, the Tokyo rainfall data and the coal mining disaster data, and by a credit--scoring problem for generalized semiparametric models. Keywords: B--spline basis; knot selection; nonnormal response...

Citations

969	Generalized Additive Models - Hastie, Tibshirani - 1990
575	Reversible jump Markov chain Monte Carlo computation and Bayesian model determination - Green - 1995
347	B: Nonparametric Regression and Generalized Linear Modes: A Roughness Penalty Approach Chapman and Hall - Green, Silverman - 1994
325	PJ: On Bayesian analysis of mixtures with an unknown number of components (with discussion - Richardson, Green - 1997
208	Tutz G: Multivariate Statistical Modelling based on Generalized Linear Models. 2 edition - Fahrmeir - 2001
190	Spline Smoothing and Nonparametric Regression - Eubank - 1988
132	Varying-coefficients models - Hastie, Tibshirani - 1993
121	Non-Gaussian state-space modeling of nonstationary time series - Kitagawa - 1987
121	Polynomial Splines and their Tensor Products in Extended Linear Modeling (with discussion - Stone, Hansen, et al. - 1997
111	Flexible Smoothing with B-splines and Penalties - Eilers, Marx - 1996
69	Flexible parsimonious smoothing and additive modeling (with discussion), Technometrics 31 - Friedman, Silverman - 1989
40	Efficient Sampling from the Posterior Distribution - Gamerman - 1997
40	Adaptive rejection Metropolis sampling within Gibbs sampling - Gilks, Best, et al. - 1995
39	Markov chain Monte Carlo methods based on “slicing” the density function - NEAL - 1997
34	Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling - Dellaportas, Smith - 1993
31	Bayesian computation and stochastic systems,” Statist - Besag, Green, et al. - 1995
25	Semi-parametric generalized linear models - Green, Yandell - 1985
24	Bayesian Deviance, the Effective Number of Parameters, and the Comparison of Arbitrarily Complex Models - SPIEGELHALTER, BEST, et al. - 1998
23	Automatic Bayesian curve - Denison, Mallick, et al. - 1998
14	Adaptive Rejection Sampling for Gibbs Sampling, Applied Statistics - Gilks, Wild - 1992
8	Automatic Bayesian curve tting - Denison, Mallick, et al. - 1998
7	Computation of Smoothing and Interpolating Natural Splines via Local Bases - Lyche, Schumaker - 1973
4	Varying-coe cient models - Hastie, Tibshirani - 1993
2	Knot Insertion for Natural Splines - Lyche, Strm - 1996
2	Spline functions: basic theory, reprinted with corrections edn - Schumaker - 1993
1	Bayesian deviance, the e ective number of parameters, and the comparison of arbitrarily complex models, Research Report 98-009 - Spiegelhalter, Best, et al. - 1998