Adaptive estimation for inverse problems with noisy operators

Cavalier, Laurent; Hengartner, Nicolas W.

doi:10.1088/0266-5611/21/4/010

Cited by 55 publications

(69 citation statements)

References 20 publications

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“…Some recent references include but are not limited to Cavalier, Golubev, Picard, and Tsybakov (2002), Cohen, Hoffmann, and Reiss (2004) and Cavalier (2008), of which density deconvolution is an important and extensively-studied problem (see, e.g., Carroll and Hall (1988); Zhang (1990);Fan (1991);Hall and Meister (2007); Lounici and Nickl (2011)). There are also papers on statistical linear ill-posed inverse problems with pseudo-unknown operators (i.e., known eigenfunctions but unknown singular values) (see, e.g., Cavalier and Hengartner (2005), Loubes and Marteau (2012)). Related papers that allow for an unknown linear operator but assume the existence of an estimator of the operator (with rate) include Efromovich and Koltchinskii (2001), Hoffmann and Reiss (2008) and others.…”

Section: Introductionmentioning

confidence: 99%

Optimal uniform convergence rates for sieve nonparametric instrumental variables regression

Chen¹,

Christensen

2013

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We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L 2 -norm rates for severely ill-posed problems, and are power of log(n) slower than the optimal L 2 -norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided.JEL Classification: C13, C14, C32

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Section: Introductionmentioning

confidence: 99%

Optimal uniform convergence rates for sieve nonparametric instrumental variables regression

Chen¹,

Christensen

2013

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“…Loubes and Ludeña (2008) also consider a setting in which A is known. Efromovich and Koltchinskii (2001), Cavalier and Hengartner (2005), Hoffmann and Reiss (2008), and Marteau (2006Marteau ( , 2009 …”

Section: Review Of Related Mathematics and Statistics Literaturementioning

confidence: 99%

Adaptive nonparametric instrumental variables estimation: empirical choice of the regularization parameter

Horowitz¹

2013

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in ABSTRACTIn nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The optimal value of this parameter depends on unknown population characteristics and cannot be calculated in applications. Theoretically justified methods for choosing the regularization parameter empirically in applications are not yet available. This paper presents such a method for use in series estimation, where the regularization parameter is the number of terms in a series approximation to g . The method does not require knowledge of the smoothness of g or of other unknown functions. It adapts to their unknown smoothness. The estimator of g based on the empirically selected regularization parameter converges in probability at a rate that is at least as fast as the asymptotically optimal rate multiplied by 1/ 2 (log ) n , where n is the sample size. The asymptotic integrated mean-square error (AIMSE) of the estimator is within a specified factor of the optimal AIMSE.

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“…For a semi-parametric setting see Butucea and Matias (2005). Recently, several statistical papers have considered adaptive estimation in the Gaussian white noise framework with a noisy kernel, see Cavalier and Hengartner (2005), Cavalier and Raimondo (2007), Efromovich and Koltchinskii (2001), Hoffmann and Reiss (2008), Marteau (2006) and Willer (2006).…”

Section: Related Work In the White Noise Modelmentioning

confidence: 99%

Multiscale density estimation with errors in variables

Cavalier

Raimondo

2010

Journal of the Korean Statistical Society

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a b s t r a c tWe present an adaptive method for density estimation when the observations X = (X 1 , . . . , X n ) are contaminated by additive errors Y = X + Z. The error distribution is not specified by the model but is estimated using repeated measurements (Y i ) i=1,2,3 of X. In this setting, we propose a wavelet method for density estimation which adapts both to the degree of ill posedness of the problem (smoothness of the error distribution) and to the regularity of the target density. Our method is implemented in the Fourier domain via a square-root transformation of the empirical characteristic function and yields fast translation invariant non-linear wavelet approximations with data-driven choices of fine tuning parameters. When the variable X is observed without errors our method provides a natural implementation of direct density estimation in the Meyer wavelet basis. We illustrate the adaptiveness properties of our estimator with a range of finite sample examples drawn from population with smooth and less smooth density functions.

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Adaptive estimation for inverse problems with noisy operators

Cited by 55 publications

References 20 publications

Optimal uniform convergence rates for sieve nonparametric instrumental variables regression

Optimal uniform convergence rates for sieve nonparametric instrumental variables regression

Adaptive nonparametric instrumental variables estimation: empirical choice of the regularization parameter

Multiscale density estimation with errors in variables

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