Statistica Sinica 19 (2009), 1389-1406

MINIMAX RATES OF CONVERGENCE AND OPTIMALITY

OF BAYES FACTOR WAVELET REGRESSION

ESTIMATORS UNDER POINTWISE RISKS

Natalia Bochkina and Theofanis Sapatinas

University of Edinburgh and University of Cyprus

Abstract: We consider function estimation in nonparametric regression over Besov spaces and under pointwise $l^u$-risks ( $1\leq u<\infty$). First we derive both non-adaptive and adaptive minimax pointwise rates of convergence in the standard nonparametric regression model, complementing recent related results obtained in the Gaussian white noise model. Then we investigate theoretical performance of Bayes factor estimators at a single point in wavelet regression models with independent and identically distributed errors that are not necessarily normally distributed. We compare both non-adaptive and adaptive Bayes factor estimators in terms of their frequentist optimality over Besov spaces and under pointwise $l^u$-risks ( $1\leq u<\infty$) for various combinations of error and prior distributions, extending recent non-adaptive results obtained for error and prior models with exponential descents and under pointwise $l^2$-risks. We provide sufficient conditions that determine whether the unknown response function belongs to a Besov space a-priori with probability one, and identify regions wherein the response function enjoys both pointwise optimality, for the proposed non-adaptive and adaptive Bayes factor estimators, and a-priori Besov membership. A simulation study is conducted to illustrate the performance of the proposed adaptive Bayes factor estimation procedure with hyperparameters estimated in a fully Bayesian framework.

Key words and phrases: Bayesian inference, Besov spaces, nonparametric regression, optimality, pointwise risk, wavelets.