# An adaptation theory for nonparametric confidence intervals

@article{Cai2004AnAT, title={An adaptation theory for nonparametric confidence intervals}, author={T. Tony Cai and Mark G. Low}, journal={Annals of Statistics}, year={2004}, volume={32}, pages={1805-1840} }

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.

## 99 Citations

Adaptive confidence intervals for regression functions under shape constraints

- Mathematics
- 2013

Adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity. A natural benchmark is established for the minimum expected length of…

A sharp adaptive confidence ball for self-similar functions

- Mathematics
- 2014

In the nonparametric Gaussian sequence space model an l2-confidence ball Cn is constructed that adapts to unknown smoothness and Sobolev-norm of the infinite-dimensional parameter to be estimated.…

On Adaptive Estimation of Linear Functionals

- Mathematics
- 2005

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in…

Adaptation under probabilistic error for estimating linear functionals

- Mathematics
- 2006

The problem of estimating linear functionals based on Gaussian observations is considered. Probabilistic error is used as a measure of accuracy and attention is focused on the construction of…

Adaptive confidence sets in $$L^2$$

- Mathematics
- 2011

The problem of constructing confidence sets that are adaptive in $$L^2$$-loss over a continuous scale of Sobolev classes of probability densities is considered. Adaptation holds, where possible, with…

Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity

- Mathematics
- 2015

Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high dimensional linear regression with random design. We first establish the…

Optimal Adaptive Inference in Random Design Binary Regression

- Mathematics
- 2015

We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density. These confidence sets are adaptive in $L^2$ loss over a continuous class of…

ON ADAPTIVE ESTIMATION OF LINEAR FUNCTIONALS1 BY T. TONY CAI

- 2005

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in…

CONFIDENCE BANDS IN DENSITY ESTIMATION

- Mathematics
- 2010

Given a sample from some unknown continuous density f: ℝ → ℝ, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of t-Holder balls, 0 < t ≤ r,…

Confidence sets for nonparametric wavelet regression

- Mathematics
- 2005

We construct nonparametric confidence sets for regression functions using wavelets that are uniform over Besov balls. We consider both thresholding and modulation estimators for the wavelet…

## References

SHOWING 1-10 OF 24 REFERENCES

On nonparametric confidence intervals

- Mathematics
- 1997

An inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage…

Adaptive confidence interval for pointwise curve estimation

- Mathematics
- 2000

We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around f(x 0 ), in either a white noise model or a regression setting. A suitable…

On Modulus of Continuity And Adaptability in Nonparametric Functional Estimation

- 2002

We study adaptive estimation of linear functionals over a collection of finitely many parameter spaces.A between class modulus of continuity, a geometric quantity, is introduced and is shown to be…

Minimax estimation of linear functionals over nonconvex parameter spaces

- Mathematics
- 2004

The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms…

Confidence sets for nonparametric wavelet regression

- Mathematics
- 2005

We construct nonparametric confidence sets for regression functions using wavelets that are uniform over Besov balls. We consider both thresholding and modulation estimators for the wavelet…

Asymptotic equivalence of nonparametric regression and white noise

- Mathematics
- 1996

The principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely.…

Statistical Estimation and Optimal Recovery

- Mathematics
- 1994

New formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random Gaussian noise. The formulas cover a variety of…

New goodness-of-fit tests and their application to nonparametric confidence sets

- Mathematics
- 1998

Suppose one observes a process V on the unit interval, where dV = f 0 +dW with an unknown parameter f 0 ∈ L 1 [0, 1] and standard Brownian motion W. We propose a particular test of one-point…

Affine minimax confidence intervals for a bounded normal mean

- Mathematics
- 1992

Consider the problem of constructing a fixed-length confidence interval for [theta]0 from the observation Y ~ N([theta]0, [sigma]2), when we know a priori that [theta]0 [epsilon][-[tau], [tau]]. The…

BOOTSTRAP SIMULTANEOUS ERROR BARS FOR NONPARAMETRIC REGRESSION

- Mathematics
- 1991

Simultaneous error bars are constructed for nonparametric kernel estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated residual…