Asymptotically Efficient Estimation of Analytic Functions in Gaussian Noise

Golubev, Yuri; Levit, B. Ya.; Tsybakov, Alexandre B.

doi:10.2307/3318549

Cited by 27 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a detailed treatment of estimating analytic functions at a point see for density estimation and Golubev, Levit and Tsybakov (1996) for nonparametric regression where particular linear estimators are constructed which are asymptotically efficient.…”

Section: Examplesmentioning

confidence: 99%

A note on nonparametric estimation of linear functionals

Cai¹,

Low²

2003

Ann. Statist.

View full text Add to dashboard Cite

show abstract

Section: Examplesmentioning

confidence: 99%

A note on nonparametric estimation of linear functionals

Cai¹,

Low²

2003

Ann. Statist.

View full text Add to dashboard Cite

show abstract

“…Such classes appeared in the statistical literature in [16], and we subsequently used in the context of density estimation [12], functional estimation, regression problems [11], and tomography [5]. In [4] we have argued that from the physical point of view it is natural to assume that the Wigner function of a state which can be created in the lab belongs to such a class.…”

Section: The Main Resultsmentioning

confidence: 99%

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

Guţǎ

Luis

2007

Math. Meth. Stat.

View full text Add to dashboard Cite

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a "quasiprobability density" on R 2 which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function, and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.

show abstract

“…The adaptive minimax rates of convergence were studied in [3,4]. As the first publications on the estimation of supersmooth functions with the sharp minimax asymptotic one should mention [10,15,16,27]. This paper can be considered as natural complement of [11].…”

Section: η(T) = H(t)ζ(t) ( 2 ) Where ζ(T) Is a Stationary Gaussian Prmentioning

confidence: 99%

Minimax and bayes estimation in deconvolution problem

Ermakov

2008

ESAIM: PS

View full text Add to dashboard Cite

Abstract.We consider a deconvolution problem of estimating a signal blurred with a random noise.The noise is assumed to be a stationary Gaussian process multiplied by a weight function function h where h ∈ L2(R 1 ) and is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [5].Mathematics Subject Classification. 62G05, 65R30, 65R32.

show abstract

Asymptotically Efficient Estimation of Analytic Functions in Gaussian Noise

Cited by 27 publications

References 0 publications

A note on nonparametric estimation of linear functionals

A note on nonparametric estimation of linear functionals

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

Minimax and bayes estimation in deconvolution problem

Contact Info

Product

Resources

About