Date of this Version
The Annals of Statistics
The focus of the present paper is on constructing l2 balls as confidence sets. There are some advantages that come with the focus on balls for confidence sets. For bands results in Low (1997) rule out the possibility of adaptation over even a pair of Lipschitz or Sobolev spaces at least for confidence bands that have a guaranteed coverage level. On the other hand, fully rate adaptive confidence balls which do maintain coverage probability can be constructed over Sobolev smoothness levels that range over an interval [α, 2α]. However, this range of models where such adaptation is possible is still quite limited and here the authors develop a theory that applies over a broader class of models. The approach taken, following Gin´e and Nickl (2010) and Bull (2012), is to focus on parameters that are in some sense typical and removing a set of parameter values that cause difficulties at least when constructing adaptive sets. The goal is then to construct fully adaptive confidence sets over the remaining collection of parameter values. In the present paper the parameter values that are kept belong to a class of parameters that they call polished tail sequences and the authors develop results that show that a particular empirical Bayes credible ball is both honest when restricted to such sequences and adaptive in size.
The original and published work is available at: https://projecteuclid.org/euclid.aos/1434546209
Low, M. G., & Ma, Z. (2015). Discussion of "Frequentist of Coverage of Adaptive Nonparametric Bayesian Credible Sets. The Annals of Statistics, 43 (4), 1448-1454. http://dx.doi.org/10.1214/15-AOS1270D
Date Posted: 27 November 2017
This document has been peer reviewed.