An adaptation theory for nonparametric confidence intervals

By Mark G. Low, Tommaso Cai. 2004

101 citations recorded by the source corpus.

A nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals. A between class modulus of continuity captures the expected length of adaptive confidence intervals. Sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds. In addition, minimax theory over nonconvex parameter spaces is developed.

Subfields: Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f7c0d28fa8cc2b9d8db37c4a7c127efed41602ce

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In Control and Systems Engineering

In Statistics and Probability

Citations in the corpus

On adaptive inference and confidence bands (2011, background): “THEOREM 1 (Low).”
CONFIDENCE BANDS IN DENSITY ESTIMATION (2010, background): “1) holds, and, following Cai and Low (2004), we shall say that Cn(y) is adaptive if for every t, ε > 0 there exists L′ finite such that [l(I) denoting the length of the interval I ] sup f∈Σ(t,L)∩P Prf (”
Optimal inference in a class of regression models (2015, methodology): “The bounds of Low (1997) and Cai and Low (2004a) imply that when F is constrained only by bounds on a derivative, the expected length of any CI that maintains coverage must shrink at the minimax rate for any any f in the interior of F . We extend and sharpen these findings by showing that, for smoot”
Impossible inference in econometrics: Theory and applications (2016, background): “It is worth highlighting the statistics literature on the impossibility of adaptation gains for confidence intervals on linear functionals of non-parametric functions (Low (1997), Cai and Low (2004)).”
Simple and honest confidence intervals in nonparametric regression (2016, background): “(4) below) that allows us to apply the “renormalization heuristics” of Donoho and Low (1992); we are thus able to cover settings such as density estimation or estimation of a bidder valuation in first-price auctions (see Supplemental Appendix C).”
Adaptive confidence sets in $$L^2$$ (2011, background): “In contrast in the context of the L∞-theory Low’s [28] by now almost classical negative result (see also [6]) has only recently been revisited: In the papers [14, 18, 23, 5], a ’separation approach’ to the problem of adaptive confidence bands was introduced, which attempts to find ’maximal’ subsets ”
Estimation and Confidence Sets for Sparse Normal Mixtures (2006, methodology): “A similar formulation for the construction of optimal nonparametric confidence intervals is given in Cai and Low [3].”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, methodology): “For example, such a construction of least favorable set has been used under the Gaussian sequence model in [1] for signal detection and in [7, 8] for estimation and confidence intervals for linear functionals.”
Confidence sets for nonparametric wavelet regression (2005, methodology): “We consider universal shrinkage [Donoho and Johnstone (1995a)], modulation estimators [Beran and Dümbgen (1998)] and a restricted form of SureShrink [Donoho and Johnstone (1995b)].”
Anti-concentration and honest, adaptive confidence bands (2013, background): “Following [6], we say the confidence band Cn is adaptive if for every t, ε > 0 there exists C > 0 such that for all n≥ 1,”
On adaptive posterior concentration rates (2013, background): “The behaviour of such a pair (ℓ, d) is reminiscent of several phenomena in the minimax nonparametric literature: these include estimation of linear functionals Donoho and Liu [15], constrained risk inequalities Brown and Low [5], Cai and Low [9], Cai et al. [8] or the existence of adaptive confidenc”
Sensitivity analysis using approximate moment condition models (2018, result): “This form of misspecification has previously been considered in a number of papers, including Hahn and Hausman (2005), Conley et al. (2012), and Andrews et al.”