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Abstract:
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We consider a semiparametric Bayesian non-linear regression model with iid errors. We do not impose a parametric form for the likelihood function; rather, we treat the true density function of error terms as an infinite dimensional nuisance parameter and estimate it nonparametrically. Once the likelihood of parameters of interest is constructed based on the estimated error density, one can conduct conventional parametric Bayesian inference using MCMC methods. We derive the asymptotic properties of the resulting estimator. In particular, we identify conditions under which our two-step Bayes estimator has the same asymptotic normal distribution that is enjoyed by the Bayes estimator that could be obtained if we knew the true density. Hence, we establish that in certain Bayesian models it is possible to obtain parallel results to the adaptive estimation literature, that is, asymptotically there is no eciency loss due to not knowing the functional form of underlying distribution.
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