Minimax convergence rates under the -risk in the functional deconvolution model

Petsa, Athanasia; Sapatinas, Theofanis

doi:10.1016/j.spl.2009.03.028

Cited by 8 publications

(6 citation statements)

References 7 publications

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“…The steps are similar to those of [14] and [3], with necessary modifications. In the supersmooth case we do not need the maxiset theorem but proceed according to [32] and consider the L p -risk (1 ≤ p < ∞) directly.…”

Section: Proofsmentioning

confidence: 99%

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Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
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13
3
8
0
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We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk (1 ≤ p < ∞). Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.

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Section: Proofsmentioning

confidence: 99%

“…where ψ is the Meyer wavelet and κ = (j , k ). The result in (32) would seem to imply that the covariance matrix of z m is non-trivial. However, applying Lemma 1, the covariance matrix reduces to…”

Section: Stochastic Analysis Of Estimated Wavelet Coefficientsmentioning

confidence: 99%

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
Self Cite
13
3
8
0
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“…In particular, Brown and Low (1996) and Brown et al (2002) in the univariate case and Reiss (2008) in the multivariate case established, under some restrictions, asymptotic equivalence (in the Le Cam sense) between nonparametric regression and Gaussian white noise models. Although, to the best of our knowledge, such an asymptotic equivalence between continuous and discrete models, in the functional deconvolution setting, has not yet been explored, it has been documented in the literature a convergence rate equivalency, in the asymptotical minimax sense, between standard continuous and discrete deconvolution models, that is, when a = b, M = 1 and N = n in (1.1) and (1.2), over a wide range of Besov balls and for the L rrisks, 1 ≤ r < ∞ [e.g., Chesneau (2008), Pensky and Sapatinas (2009a) and Petsa and Sapatinas (2009)].…”

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confidence: 99%

On convergence rates equivalency and sampling strategies in functional deconvolution models

Pensky¹,

Sapatinas²

2010

Ann. Statist.

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Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its real-life discrete counterpart over a wide range of Besov balls and for the L 2 -risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these models irregular.We formulate the conditions when each of these situations takes place. In the regular case, we not only point out the number and the selection of sampling points which deliver the fastest convergence rates in the discrete model but also investigate when, in the case of an arbitrary sampling scheme, the convergence rates in the continuous model coincide or do not coincide with the convergence rates in the discrete model. We also study what happens if one chooses a uniform, or a more general pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement of the continuous model. Finally, as a representative of the irregular case, we study functional deconvolution with a boxcar-like blurring function since this model

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“…See e.g. Cai (1999), Cai (2002), Pensky and Sapatinas (2009), Petsa and Sapatinas (2009) and Chesneau et al (2010).…”

Section: Minimax Resultsmentioning

confidence: 99%

Wavelet density estimators for the deconvolution of a component from a mixture

Chesneau

2011

Sankhya A

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We consider the model: Y = X + , where X and are independent random variables. The density of is known whereas the one of X is a finite mixture with unknown components. Considering the "ordinary smooth case" on the density of , we want to estimate a component of this mixture. To reach this goal, we develop two wavelet estimators: a nonadaptive based on a projection and an adaptive based on a hard thresholding rule. We evaluate their performances by taking the minimax approach under the mean integrated squared error over Besov balls. We prove that the adaptive one attains a sharp rate of convergence.

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Minimax convergence rates under the -risk in the functional deconvolution model

Cited by 8 publications

References 7 publications

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
Self Cite
13
3
8
0
View full text Add to dashboard Cite

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
Self Cite
13
3
8
0
View full text Add to dashboard Cite

On convergence rates equivalency and sampling strategies in functional deconvolution models

Wavelet density estimators for the deconvolution of a component from a mixture

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Minimax convergence rates under the -risk in the functional deconvolution model

Cited by 8 publications

References 7 publications

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞Kulik1, Sapatinas2, Wishart3 2015Applied and Computational Harmonic AnalysisSelf Cite13380View full textAdd to dashboardCite

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞Kulik1, Sapatinas2, Wishart3 2015Applied and Computational Harmonic AnalysisSelf Cite13380View full textAdd to dashboardCite

On convergence rates equivalency and sampling strategies in functional deconvolution models

Wavelet density estimators for the deconvolution of a component from a mixture

Contact Info

Product

Resources

About

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
Self Cite
13
3
8
0
View full text Add to dashboard Cite

Multichannel deconvolution with long range dependence: Upper bounds on theLp-risk(1≤p<∞
Kulik
¹
,
Sapatinas
²
,
Wishart
³

2015
Applied and Computational Harmonic Analysis
Self Cite
13
3
8
0
View full text Add to dashboard Cite