BERNSTEIN-VON MISES THEOREMS FOR GAUSSIAN REGRESSION WITH INCREASING NUMBER OF REGRESSORS
BERNSTEIN-VON MISES THEOREMS FOR GAUSSIAN REGRESSION WITH INCREASING NUMBER OF REGRESSORS
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DOI：
10.1214/11-aos912
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发表时间：
2011-10-01
影响因子：
4.5
通讯作者：
Bontemps, Dominique
中科院分区：
文献类型：
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作者：
Bontemps, Dominique
本文对非参数和半参数估计的贝叶斯理论做出了贡献。当回归量的数量随着样本大小的增加而增加时，我们对高斯线性回归模型中后验分布的渐近正态性感兴趣。在此框架中获得了两种 Bernstein-von Mises 定理：参数本身的非参数定理和参数泛函的半参数定理。我们将它们应用于高斯序列模型以及 Sobolev 和 C-alpha 类中的函数回归，其中我们得到了极小极大收敛率。我们的应用程序中泛函的贝叶斯估计量达到了自适应性。
This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and C-alpha classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.