Adaptive nonparametric confidence sets

By James M. Robins, A. van der Vaart. 2006

109 citations recorded by the source corpus.

We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models. The latter adaptation is necessarily limited in scope. We review the notion of adaptive confidence regions, and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation. Applications include the finite normal mean model, the white noise model, density estimation and regression with random design. © Institute of Mathematical Statistics, 2006.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3568de0fd69b56e60e5087fbffcbcff2ae163a64

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Citations in the corpus

Uniform asymptotic inference and the bootstrap after model selection (2015, background): “k, and the proof that the partition elements ΠM , M ∈ M are cones for LAR and lasso, mirrors that in Taylor et al. (2014). Like FS, the active sets from LAR are nested, Â1(y)⊆ Â2(y)⊆ Â3(y)⊆ .”
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).”
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).”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, background): “See, for example, [6] for adaptive confidence intervals for linear functionals, [13, 9] for adaptive confidence bands, and [8, 20] for adaptive confidence balls.”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, background): “Similar restrictions are valid for honesty over subsets of R, as follows from testing arguments (see the appendix in [33]).”
Adaptive Confidence Balls (2006, background): “Robins and van der Vaart [12] use sample splitting to divide the construction of the center and radius of a confidence ball into independent problems and show how to use estimates of quadratic functionals to construct adaptive confidence balls.”