ABSTRACT 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.

中文翻译：

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具有未知误差分布的非线性回归模型的可信区间的构建

摘要 贝叶斯回归一直引起人们的兴趣，在贝叶斯回归中不对误差分布施加参数假设。在这项研究中，我们考虑半参数贝叶斯非线性回归模型。我们不对似然函数强加参数形式。相反，我们将误差项的真实密度函数视为一个无限维的干扰参数，并以非参数方式对其进行估计。此后，我们使用 MCMC 方法进行传统的参数贝叶斯推理。我们推导出所得估计器的渐近特性并确定自适应估计的条件，在该条件下，我们的两步贝叶斯估计器享有与我们知道真实密度相同的渐近效率。我们使用模拟和真实数据，根据经验将自适应贝叶斯点和区间估计量与最大似然估计量的准确性和覆盖率进行比较。特别是，我们观察到贝叶斯推理可能在小样本量的数值稳定性方面更胜一筹。