Adaptive Confidence Balls

By Tommaso Cai, Mark G. Low. 2006

77 citations recorded by the source corpus.

Adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework. Finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level. The confidence balls are centered on adaptive estimators based on special local block thresholding rules. The radius is derived from an analysis of the loss of this adaptive estimator. In addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of R N and expected squared radius adapting over a maximum range of Besov bodies.

Subfields: Computer Networks and Communications, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/43edefa40719b44018c3d51e9a8415446925e6ef

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In Computer Networks and Communications

In Finance

In Statistics and Probability

Citations in the corpus

Confidence sets in sparse regression (2012, methodology): “In nonparametric estimation problems this has been shown to be possible; see Hoffmann and Lepski (2002), Juditsky and Lambert-Lacroix (2003), Cai and Low (2006), Robins and van der Vaart (2006), Bull and Nickl (2013). Translated to high-dimensional linear models, the accuracy of these methods can be”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, background): “It is known, see for instance (Cai and Low, 2006) that there does not exist honest confidence regions (i.e. satisfying (12)) which have adaptive size, unless A ⊂ [β1, 2β1] for some β1.”
On adaptive inference and confidence bands (2011, background): “It was already noted or implicitly used in [1, 5, 15, 19, 29] that there is a generic connection between adaptive confidence sets and minimax distinguishability of certain nonparametric hypotheses.”
Adaptive confidence sets in $$L^2$$ (2011, background): “A first result, the key elements of which have been discovered and discussed in [6, 16,21,24,28], is that L2-adaptive confidence statements that parallel the situation of Theorem 2 exist without any additional restrictions whatsoever, in the case where R < 2r , so that the window of adaptation is [r”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “The issue of constructing nonparametric confidence set has been studied by many authors, see [11, 1, 16].”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, background): “18 in [9] and [6] and the second inequality follows from (2.”
CONFIDENCE BANDS IN DENSITY ESTIMATION (2010, unknown): “We refer to Li (1989), Beran and Dümbgen (1998), Hoffman and Lepski (2002), Juditsky and Lambert-Lacroix (2003), Baraud (2004), Genovese and Wasserman (2005), Cai and Low (2006) and Robins and van der Vaart (2006).”