Construction of credible intervals for nonlinear regression models with unknown error distributions

There has been continuing interest in Bayesian regressions in which no parametric assumptions are imposed on the error distribution. In this study, we consider semiparametric Bayesian nonlinear regression models. We do not impose a parametric form for the likelihood function. Instead, we treat the true density function of error terms as an infinite-dimensional nuisance parameter and estimate it nonparametrically. Thereafter, we conduct a conventional parametric Bayesian inference using MCMC methods. We derive the asymptotic properties of the resulting estimator and identify conditions for adaptive estimation, under which our two-step Bayes estimator enjoys the same asymptotic efficiency as if we knew the true density. We compare the accuracy and coverage of the adaptive Bayesian point and interval estimators to those of the maximum likelihood estimator empirically, using simulated and real data. In particular, we observe that the Bayesian inference may be superior in numerical stability for small sample sizes.