Bayesian wavelet estimation of partially linear models

Abstract

A Bayesian wavelet approach is presented for estimating a partially linear model (PLM). A PLM consists of a linear part and a nonparametric component. The nonparametric component is represented with a wavelet series where the wavelet coefficients have assumed prior distributions. The prior for each coefficient consists of a mixture of a normal distribution and a point mass at 0. The linear parameters are assumed to have a normal prior. The hyperparameters are estimated by the marginal maximum likelihood estimator using the direct maximization. The model selection and model averaging methods give different estimates of the model parameters. MCMC computation is used for the estimation of the linear coefficients by model averaging method. Simulated examples illustrate the performance of the proposed estimators.

Keywords:

• Partially linear models
• Bayesian
• Discrete wavelet transform
• Mixture distribution
• MCMC
• Model selection
• Model averaging

Acknowledgements

Part of the work is part of the author's Ph.D dissertation at the Statistics Department of Purdue University, under the supervision of Professor Mary Ellen Bock, whose generous guidance and consistent encouragement are gratefully acknowledged and appreciated. I am grateful to an anonymous referee for comments that led to a much improved manuscript. Thanks go to Professor Peter J. Lenk for providing the traffic accident data set.