Density Estimation for the Case of Supersmooth Measurement Error

Efromovich, Sam

doi:10.2307/2965701

Cited by 50 publications

(66 citation statements)

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“…Therefore in this case adaptation can be achieved completely for free. Examples of such a phenomenon in the context of density estimation with supersmooth measurement error have been given in Efromovich (1997a). Also see Efromovich (1997b) and Efromovich and Koltchinskii (2001) for further examples in the context of other inverse problems.…”

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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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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“…Although it is standard in the statistic literature to assume that g is known, it may not very realistic in practice. In some circumstances, there exists an auxiliary data set that can be used to estimate parametrically or nonparametrically the function g. In for instance Efromovich (1997), Johannes (in press), and Neumann (2007), g is estimated nonparametrically. In our application in Section 6, we postulate a parametric form for g and estimate the unknown parameters using an auxiliary sample.…”

Section: Estimation Of Tmentioning

confidence: 99%

A Spectral Method for Deconvolving a Density

Carrasco

Florens

2010

Econom. Theory

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Abstract:We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and the symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the problem is identi…ed even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.2

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“…Under this assumption, rates of convergence and their optimality for kernel estimators have been studied in Carroll and Hall (1988), Stefanski (1990), Stefanski and Carroll (1990), Fan (1991) and Efromovich (1997). For the study of sharp asymptotic optimality, we can cite Butucea (2004), Butucea and Tsybakov (2008a,b).…”

Section: Bibliography For Real-valued Variablesmentioning

confidence: 99%

Laguerre deconvolution with unknown matrix operator

Comte

Mabon

2017

Math. Meth. Stat.

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Abstract. In this paper we consider the convolution model Z = X + Y with X of unknown density f , independent of Y , when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent, identically, distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in Mabon (2016b). To make the problem identifiable for unknown density of Y , we assume that we have access to a preliminary sample of the nuisance process Y . The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, particularly adapted to the non-negativeness of both random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein's type concentration inequalities developed for random matrices in Tropp (2015). Finally we illustrate our method on some simulated data.

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Density Estimation for the Case of Supersmooth Measurement Error

Cited by 50 publications

References 0 publications

A note on nonparametric estimation of linear functionals

A note on nonparametric estimation of linear functionals

A Spectral Method for Deconvolving a Density

Laguerre deconvolution with unknown matrix operator

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