Critical dimension in the semiparametric Bernstein—von Mises theorem

The classical parametric and semiparametric Bernstein-von Mises (BvM) results are reconsidered in a nonclassical setup allowing finite samples and model misspecification. In the parametric case and in the case of a finite-dimensional nuisance parameter, we establish an upper bound on the error of Ga...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the Steklov Institute of Mathematics 2014-12, Vol.287 (1), p.232-255
Main Authors: Panov, Maxim E., Spokoiny, Vladimir G.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The classical parametric and semiparametric Bernstein-von Mises (BvM) results are reconsidered in a nonclassical setup allowing finite samples and model misspecification. In the parametric case and in the case of a finite-dimensional nuisance parameter, we establish an upper bound on the error of Gaussian approximation of the posterior distribution of the target parameter; the bound depends explicitly on the dimension of the full and target parameters and on the sample size. This helps to identify the so-called critical dimension p n of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition “ p n 3 / n is small” is sufficient for the BvM result to be valid under general assumptions on the model. We also provide an example of a model with the phase transition effect: the statement of the BvM theorem fails when the dimension p n approaches n 1/3 .
ISSN:0081-5438
1531-8605
DOI:10.1134/S0081543814080148