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A Bernstein-von Mises theorem for smooth functionals in semiparametric models. (English) Zbl 1327.62302
Summary: A Bernstein-von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared \(L^{2}\)-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.

MSC:
62G20 Asymptotic properties of nonparametric inference
62M15 Inference from stochastic processes and spectral analysis
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