On Nonparametric Estimation of the Value of a Linear Functional in Gaussian White Noise

Ибрагимов, И. А.; Khas’minskiĭ, R. Z.

doi:10.1137/1129002

Cited by 113 publications

(106 citation statements)

References 4 publications

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“…In this setup Ibragimov and Hasminskii (1984), Donoho and Liu (1991) and Donoho (1994) give a detailed study of the minimax affine risk R A . In particular, asymptotic descriptions of the minimax risk are given when the modulus is Hölderian.…”

Section: Minimax Theorymentioning

confidence: 99%

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A note on nonparametric estimation of linear functionals

Cai¹,

Low²

2003

Ann. Statist.

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Section: Minimax Theorymentioning

confidence: 99%

“…For estimating a linear functional Ibragimov and Hasminskii (1984) described the linear estimator which has smallest maximum mean squared error assuming that F is convex and symmetric. Donoho and Liu (1991) and Donoho (1994) extended this theory to the case where F is assumed convex but need not be symmetric.…”

Section: Introduction We Observe Data Y Of the Formmentioning

confidence: 99%

A note on nonparametric estimation of linear functionals

Cai¹,

Low²

2003

Ann. Statist.

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“…standard normal random variables and M is a finite or countably infinite index set. In particular, minimax estimation theory has been well developed in Ibragimov and Hasminskii (1984), Donoho and Liu (1991) and Donoho (1994).…”

mentioning

confidence: 99%

An adaptation theory for nonparametric confidence intervals

Cai¹,

Low²

2004

Ann. Statist.

116

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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.

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“…Here T n ϭ ͕t i ͖ n iϭ1 is the design. This definition of error is commonly used in the statistical literature, see e.g., Sacks and Ylvisaker (1978), Speckman (1979), Ibragimov and Hasminski (1984), Nussbaum (1985), Donoho (1994), Donoho et al (1995).…”

Section: Non-sequential and Sequential Designsmentioning

confidence: 99%

How to Benefit from Noise

Plaskota

1996

Journal of Complexity

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We compare sequential and non-sequential designs for estimating linear functionals in the statistical setting, where experimental observations are contaminated by random noise. It is known that sequential designs are no better in the worst case setting for convex and symmetric classes, as well as in the average case setting with Gaussian distributions.In the statistical setting the opposite is true. That is, sequential designs can be significantly better. Moreover, by using sequential designs one can obtain much better estimators for noisy data than for exact data. In this way, problems that are computationally intractable for exact data may become tractable for noisy data. These results hold because adaptive observations and noise make it possible to simulate Monte Carlo.

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On Nonparametric Estimation of the Value of a Linear Functional in Gaussian White Noise

Cited by 113 publications

References 4 publications

A note on nonparametric estimation of linear functionals

A note on nonparametric estimation of linear functionals

An adaptation theory for nonparametric confidence intervals

How to Benefit from Noise

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