Bayesian quantile regression for single-index models
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Using an asymmetric Laplace distribution, which provides a mechanism for Bayesian inference of quantile regression models, we develop a fully Bayesian approach to fitting single-index models in conditional quantile regression. In this work, we use a Gaussian process prior for the unknown nonparametric link function and a Laplace distribution on the index vector, with the latter motivated by the recent popularity of the Bayesian lasso idea. We design a Markov chain Monte Carlo algorithm for posterior inference. Careful consideration of the singularity of the kernel matrix, and tractability of some of the full conditional distributions leads to a partially collapsed approach where the nonparametric link function is integrated out in some of the sampling steps. Our simulations demonstrate the superior performance of the Bayesian method versus the frequentist approach. The method is further illustrated by an application to the hurricane data.
KeywordsGaussian process prior Markov chain Monte Carlo Quantile regression Single-index models
We thank the Associate Editor and three anonymous referees for their helpful comments that have led to a significant improvement of the manuscript. Robert Gramacy would like to thank the Kemper Family Foundation for their support. The research of Heng Lian is supported by Singapore Ministry of Education Tier 1 Grant.
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