Summary
We discuss a Bayesian approach to nonparametric regression which is robust against outliers and discontinuities in the underlying function. Our approach uses Markov chain Monte Carlo methods to perform a Bayesian analysis of conditionally Gaussian state space models. In these models, the observation and state transition errors are assumed to be mixtures of normals, so the model is Gaussian conditionally on the mixture indicator variables. We present several examples of conditionally Gaussian state space models, and, for each example, we discuss several possible Markov chain Monte Carlo sampling schemes. We show empirically that our approach (i) provides a good estimate of the smooth part of the regression curve; (ii) discriminates between real and spurious jumps; and (iii) allows for outliers in the observation errors. We also show empirically that our sampling schemes converge rapidly to the posterior distribution.
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References
Carlin, B. P., Poison, N. G. and Stoffer, D. S. (1992). A Monte Carlo approach to nonnormal and nonlinear state space modelling. Journal of the American Statistical Association, 75, 493–500.
Carter, C.K. (1993). On Markov chain Monte Carlo methods for state space modelling. Unpublished PhD thesis, University of New South Wales.
Carter, C. K. and Kohn, R. (1994a). On Gibbs sampling for state space models. Biometrika, 81, 541–553.
Carter, C. K. and Kohn, R. (1994b). Markov chain Monte Carlo in conditionally Gaussian state space models. Biometrika, to appear.
Carter, C. K. and Kohn, R. (1994c). Bayesian methods for conditionally Gaussian state space models. Preprint.
De Jong, P. and Shephard, N. (1994). Efficient sampling from the smoothing density in time series models. Biometrika, to appear.
Friihwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. Journal of Time Series Analysis, 15, 183–202.
Hastie, T. J. and Tibshirani, R. J. (1990). Generalized additive models. New York: Chapman and Hall.
Kohn, R. and Ansley, C. F. (1987). A new algorithm for spline smoothing based on smoothing a stochastic process. SIAM Journal of Scientific and Statistical Computing, 8, 33–48.
Kohn, R., Ansley, C. F. and Tharm, D. (1991). The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. Journal of the American Statistical Association, 86, 1042–1050.
Lipsey, R. G., Sparks, G. R. and Steiner, P. O. (1976). Economics (2nd ed.) New York: Harper and Row.
Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika, 81, 27–40.
Marion, J. B. (1970). Classical dynamics of particles and systems. (2nd ed.) New York: Academic Press.
Meinhold, R.J. and Singpurwalla, N.D. (1989). Robustification of Kalman filter models. Journal of the American Statistical Association, 84, 479–486.
McDonald, J. A. and Owen, A. B. (1986). Smoothing with split linear fits. Technometrics, 28, 195–208.
Müller, H. G. (1992). Change-points in nonparametric regression analysis. Annals of Statistics, 20, 737–761.
Shephard, N. (1994). Partial non-Gaussian state space.Biometrika, 81, 115–132.
Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, to appear.
Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. Journal of the Royal Statistical Society, Ser. B, 40, 364–372.
Wahba, G. (1983). Bayesian “confidence intervals” for the cross-validated smoothing spline.Journal of the Royal Statistical Society, Ser. B, 45, 133–150.
Wecker, W. E. and Ansley, C. F. (1983). The signal extraction approach to nonlinear regression and spline smoothing. Journal of the American Statistical Association, 78, 81–89.
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© 1996 Physica-Verlag Heidelberg
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Carter, C.K., Kohn, R. (1996). Robust Bayesian Nonparametric Regression. In: Härdle, W., Schimek, M.G. (eds) Statistical Theory and Computational Aspects of Smoothing. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-48425-4_11
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DOI: https://doi.org/10.1007/978-3-642-48425-4_11
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-0930-5
Online ISBN: 978-3-642-48425-4
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