Honest adaptive confidence bands and self-similar functions

By Adam D. Bull. 2011

69 citations recorded by the source corpus.

Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estima- tor. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of Holder classes. While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions. We then consider the self-similar functions, whose Holder norm is similar at large and small scales. We show that such functions may be considered typical functions of a given Holder class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any Holder norm.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a79b0c41adfae38ec6338932c15d51fcccc5fa55

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Citations in the corpus

Anti-concentration and honest, adaptive confidence bands (2013, methodology): “For the density estimation problem, if Ln is a singleton, that is, the smoothing parameter is chosen deterministically, the existence of such a continuous limit distribution, which is typically a Gumbel distribution, has been established for convolution kernel density estimators and some wavelet pro”
Simple and honest confidence intervals in nonparametric regression (2016, unknown): “Bull (2012) and Chernozhukov et al. (2014) have obtained rate optimal confidence bands under such conditions, which, like the CIs considered here, use a critical value based on an upper bound on the bias.”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “…condition plays the same role as the assumptions of “self-similarity” or “polished tail” used by Picard and Tribouley (2000); Giné and Nickl (2010); Bull (2012); Nickl and Szabo (2016); Szabó et al. (2015b); Sniekers and van der Vaart (2015c); Rousseau and Szabo (2016) in their…”
Adaptive confidence sets in $$L^2$$ (2011, background): “As the results [5,13,22] prove adaptation in L∞, they naturally imply adaptation also in L2; the functions excluded, however, are now those whose norm is hard to estimate, rather than those whose norm is merely large.”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, methodology): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”