## Adaptive Bayesian Regression Splines in Semiparametric Generalized Linear Models (1998)

Venue: | Journal of Computational and Graphical Statistics |

Citations: | 23 - 2 self |

### BibTeX

@ARTICLE{Biller98adaptivebayesian,

author = {Clemens Biller},

title = {Adaptive Bayesian Regression Splines in Semiparametric Generalized Linear Models},

journal = {Journal of Computational and Graphical Statistics},

year = {1998},

volume = {9},

pages = {122--140}

}

### Years of Citing Articles

### OpenURL

### 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

1696 | Generalized Additive Models
- Hastie, Tibshirani
- 1990
(Show Context)
Citation Context ...., in Schumaker (1993) or Lyche and Str��m (1996). Further, due to the Bayesian approach using Markov chain Monte Carlo methods, extensions of the semiparametric model to generalized additive mode=-=ls (Hastie and Tibshirani, 1990-=-) or the more general varying--coefficient models (Hastie and Tibshirani, 1993) are possible without much problems. With regard to Markov chain Monte Carlo methods, other approaches for updating fixed... |

935 | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination
- Green
- 1995
(Show Context)
Citation Context ...ile in the backward steps they delete knots yielding the model being optimal for the generalized crossvalidation score. A Bayesian approach using reversible jump Markov chain Monte Carlo (RJMCMC, see =-=Green, 1995-=-) is presented by Denison, Mallick and Smith (1998). In each iteration they choose the set of knots by RJMCMC methods, and given these knots the spline is estimated by the usual least squares approach... |

542 | Nonparametric regression and generalized linear models: a roughness penalty approach - Green, Silverman - 1994 |

485 | On Bayesian analysis of Mixtures with an Unknown Number of Components - Richardson, Green - 1997 |

347 |
Multivariate Statistical Modelling Based on Generalized Linear Models
- Fahrmeir, Tutz
- 1994
(Show Context)
Citation Context ...proach to adaptive regression splines with three examples. The first two are data from the literature for curve estimation with discrete response, the Tokyo rainfall data (see e.g. Kitagawa, 1987, or =-=Fahrmeir and Tutz, 1997-=-) and the coal mining disaster data (see e.g. Eilers and Marx, 1996). The third example is an application of the semiparametric model to credit--scoring data described in Fahrmeir and Tutz (1997). 4.1... |

262 |
Spline smoothing and nonparametric regression
- Eubank
- 1988
(Show Context)
Citation Context ...Placement of a knot in a certain aera yields more flexibility of f in that aera. Since finding the right number and location of knots by visual inspection of the data is impossible in most cases (see =-=Eubank, 1988-=-, Section 7.2), we need data driven methods for knot placement to get (in some sense) nearly optimal estimators f . For normal response y, such data driven methods exist. Friedman and Silverman (1989)... |

236 | Flexible smoothing with B-splines and penalties (with discussion
- Eilers, Marx
- 1996
(Show Context)
Citation Context ...two are data from the literature for curve estimation with discrete response, the Tokyo rainfall data (see e.g. Kitagawa, 1987, or Fahrmeir and Tutz, 1997) and the coal mining disaster data (see e.g. =-=Eilers and Marx, 1996-=-). The third example is an application of the semiparametric model to credit--scoring data described in Fahrmeir and Tutz (1997). 4.1 Rainfall data The response is given by the number of occurrences o... |

225 | Varying-Coefficient Models
- Hastie, Tibshirani
- 1993
(Show Context)
Citation Context ...n approach using Markov chain Monte Carlo methods, extensions of the semiparametric model to generalized additive models (Hastie and Tibshirani, 1990) or the more general varying--coefficient models (=-=Hastie and Tibshirani, 1993-=-) are possible without much problems. With regard to Markov chain Monte Carlo methods, other approaches for updating fixed effects in the generalized linear model (Section 3.2) will be considered, sin... |

163 |
Non-Gaussian State-Space Modeling of Nonstationary Time Series
- Kitagawa
- 1987
(Show Context)
Citation Context ...ate the Bayesian approach to adaptive regression splines with three examples. The first two are data from the literature for curve estimation with discrete response, the Tokyo rainfall data (see e.g. =-=Kitagawa, 1987-=-, or Fahrmeir and Tutz, 1997) and the coal mining disaster data (see e.g. Eilers and Marx, 1996). The third example is an application of the semiparametric model to credit--scoring data described in F... |

158 | The use of polynomial splines and their tensor products in multivariate function estimation (with discussion),” The Annals of Statistics - Stone - 1994 |

93 | Flexible parsimonious smoothing and additive modeling (with discussion - Friedman, Silverman - 1989 |

80 |
Sampling from the Posterior Distribution
- Gamerman
- 1998
(Show Context)
Citation Context ...) depends only on k and k max . A widely used prior for the coefficients b = (c 0 ; fi 0 ) 0 of a generalized linear model is the multivariate normal distribution bjk �� NK+p (0; \Sigma 0 ) (see e=-=.g. Gamerman, 1997-=-). While the basis coefficients c are assumed to be uncorrelated, possible correlations between the coefficients fi = (fi 1 ; : : : ; fi p ) 0 are modelled by defining \Sigma 0 = oe 2 0 diag(I K ; R p... |

78 | Adaptive rejection Metropolis sampling within Gibbs sampling - Gilks, Best, et al. - 1995 |

52 | Markov Chain Monte Carlo methods based on ŞslicingŤ the density function - Neal - 1997 |

44 | Bayesian computation and stochastic systems,” Statist - Besag, Green, et al. - 1995 |

41 | Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling - Dellaportas, Smith - 1993 |

35 | Bayesian deviance, the effective number of parameters, and the comparison of arbitrarily complex models - Spiegelhalter, Best, et al. - 1998 |

31 | Automatic Bayesian curve - MIYATA, Denison, et al. - 1998 |

31 | The theory of generalized linear models - Gilchrist, Green - 1994 |

22 | Adaptive rejection sampling for Gibbs sampling.” Applied Statistics - Gilks, Wild - 1992 |

8 | Computation of smoothing and interpolating natural splines via local bases - Lyche, Schumaker - 1973 |

8 | Automatic Bayesian curve tting - Denison, Mallick, et al. - 1998 |

6 |
Varying-coe¢ cient models
- Hastie, Tibshirani
(Show Context)
Citation Context ...sian approach using Markov chain Monte Carlo methods, extensions of the semiparametric model to generalized additive models (Hastie and Tibshirani, 1990) or the more general varying{coe cient models (=-=Hastie and Tibshirani, 1993-=-) are possible without much problems. With regard to Markov chain Monte Carlo methods, other approaches for updating xed e ects in the generalized linear model (Section 3.2) will be considered, since ... |

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 |