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Cai, T. Tony; Low, Mark G.

An adaptation theory for nonparametric confidence intervals. (English) Zbl 1056.62060

Ann. Stat. 32, No. 5, 1805-1840 (2004).
From the paper: The problem of estimating a linear functional occupies a central position in nonparametric function estimation. It is most complete in the Gaussian settings: \[ dY(t)=f(t)dt+n^{-1/2}dW(t),\quad -1/2\leq t\leq 1/2, \] where \(W(t)\) is standard Brownian motion and \[ Y(i)=f(i)+n^{-1/2}z_i, \quad i\in{\mathcal M}, \] where \(z_i\) are i.i.d. standard normal random variables and \({\mathcal M}\) is a finite or countably infinite index set. Confidence sets also play a fundamental role in statistical inference. In the context of nonparametric function estimation variable size confidence intervals, bands and balls have received particular attention recently. For any confidence set there are two main interrelated issues which need to be considered together, coverage probability and the expected size of the confidence set.
Here, 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.

Cited in 47 Documents

MSC:

62G15 Nonparametric tolerance and confidence regions
62C20 Minimax procedures in statistical decision theory
62F35 Robustness and adaptive procedures (parametric inference)
62M99 Inference from stochastic processes

Keywords:

adaptation; between class modulus; confidence intervals; coverage; expected length; linear functionals; minimax estimation; modulus of continuity; white noise model
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References:

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