Richard Nickl

On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures

We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein-von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donsker- and Kolmogorov-Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.

UNIFORM LIMIT THEOREMS FOR WAVELET DENSITY ESTIMATORS

Let pn(y) = Σ k Φ (y-k) + Σ jn-1 l=0 Σ k lk 2l/2 ψ(2 l y-k) be the linear wavelet density estimator, where φ, ψ are a father and a mother wavelet (with compact support ? ??? are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p 0 on R, and j n ∈ Z, j n ∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup y∈R | Pn (y) - Ep n (y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup y∈R |p n (y) - po (y)| Attains the optimal almost sure rate of convergence for estimating p 0 , if j n is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of p n , that is, for the stochastic processes n(F W n (s) - F (s)) n∫ s - ∞ (pn - p 0 ), s ∈ R, are proved; and more generally, uniform central limit theorems for the processes n∫(p n - P 0 )?, f ∈ F, for other Donsker classes F of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508-539].

Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking

A simple adaptive estimator of the integrated square of a density

L^r

Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds

-norm neighborhoods,

Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

1\le r\le\infty

High-frequency Donsker theorems for Lévy measures

, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for

Spatially adaptive density estimation by localised Haar projections

1\le r\le2

Efficient simulation-based minimum distance estimation and indirect inference

, but deteriorate as

Inference on covariance operators via concentration inequalities: k-sample tests, classification, and clustering via Rademacher complexities

r