TY - JOUR

T1 - Critical dimension in the semiparametric Bernstein—von Mises theorem

AU - Panov, Maxim E.

AU - Spokoiny, Vladimir G.

PY - 2014/11/27

Y1 - 2014/11/27

N2 - 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 pn of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition “pn3/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 pn approaches n1/3.

AB - 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 pn of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition “pn3/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 pn approaches n1/3.

UR - http://www.scopus.com/inward/record.url?scp=84921899704&partnerID=8YFLogxK

U2 - 10.1134/S0081543814080148

DO - 10.1134/S0081543814080148

M3 - Article

AN - SCOPUS:84921899704

VL - 287

SP - 232

EP - 255

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - 1

ER -