On √(n)-consistency for Bayesian quantile regression based on the misspecified asymmetric Laplace likelihood
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The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al. (2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for √(n)-consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for n^α rate for α<1/2. However, √(n)-rate is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we give sufficient conditions for √(n)-consistency in the more general setting of Bayesian non-linear quantile regression based on ALD. In particular, we derive √(n)-consistency for the Bayesian linear quantile regression. Our approach also enables an interesting extension of the linear case when number of parameters p increases with n, where we obtain posterior consistency at the rate n^α for α<1/2.
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