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Dive into the research topics where Jan Johannes is active.

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Featured researches published by Jan Johannes.


Annals of Statistics | 2009

Deconvolution with unknown error distribution

Jan Johannes

We consider the problem of estimating a density fX using a sample Y1, …, Yn from fY=fX⋆fe, where fe is an unknown density. We assume that an additional sample e1, …, em from fe is observed. Estimators of fX and its derivatives are constructed by using nonparametric estimators of fY and fe and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density fe, where it is assumed that fX satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density fX belongs to a Sobolev space


Econometrics Journal | 2012

Instrumental Regression in Partially Linear Models

Jean-Pierre Florens; Jan Johannes; Sébastien Van Bellegem

H_{\mathcal{p}}


Econometric Theory | 2011

Identification and estimation by penalization in nonparametric instrumental regression

Jean-Pierre Florens; Jan Johannes; Sébastien Van Bellegem

and fe is ordinary smooth or supersmooth.


Journal of Multivariate Analysis | 2010

Thresholding projection estimators in functional linear models

Hervé Cardot; Jan Johannes

We consider the semiparametric regression Xtβ + φ (Z) where β and φ(.) are unknown slope coefficient vector and function, and where the variables (X, Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parametrization of φ generally leads to an inconsistent estimator of β, whereas consistent nonparametric estimators for β have a slow rate of convergence. An additional complication is that the solution of the equation necessitates the inversion of a compact operator which can be estimated nonparametrically. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √n -consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition which we explicit and interpret in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroskedastic.


Annals of Statistics | 2012

Adaptive functional linear regression

Fabienne Comte; Jan Johannes

The nonparametric estimation of a regression function from conditional moment restrictions involving instrumental variables is considered. The rate of convergence of penalized estimators is studied in the case where the regression function is not identified from the conditional moment restriction. We also study the gain of modifying the penalty in the estimation, considering derivatives in the penalty. We analyze the effect of this modification on the identification of the regression function and the rate of convergence of its estimator.


Journal of Econometrics | 2014

Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression

Fabian Dunker; Jean-Pierre Florens; Thorsten Hohage; Jan Johannes; Enno Mammen

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.


Bernoulli | 2013

Adaptive circular deconvolution by model selection under unknown error distribution

Jan Johannes; Maik Schwarz

We consider the estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. Cardot and Johannes [J. Multivariate Anal. 101 (2010) 395–408] have shown that a thresholded projection estimator can attain up to a constant minimax-rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives. This estimation procedure, however, requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As this information is usually inaccessible in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski’s method. It is inspired by the recent work of Goldenshluger and Lepski [Ann. Statist. 39 (2011) 1608–1632]. The tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating Lepski’s method among a random collection of admissible values. This choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data-driven thresholded projection estimator can attain minimaxrates up to a constant over a variety of classes of slope functions and covariance operators. The results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives. A simulation study shows the reasonable performance of the fully data-driven estimation procedure.


Econometric Theory | 2016

Adaptive estimation of functionals in nonparametric instrumental regression

Christoph Breunig; Jan Johannes

This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model. This is illustrated in computed examples with simulated data.


Communications in Statistics-theory and Methods | 2013

Adaptive Gaussian Inverse Regression with Partially Unknown Operator

Jan Johannes; Maik Schwarz

We consider a circular deconvolution problem, where the density f of a circular random variable X has to be estimated nonparametrically based on an iid. sample from a noisy observation Y of X. The additive measurement error is supposed to be independent of X. The objective of this paper is the construction of a fully data-driven estimation procedure when the error density φ is unknown. However, we suppose that in addition to the iid. sample from Y , we have at our disposal an additional iid. sample independently drawn from the error distribution. First, we develop a minimax theory in terms of both sample sizes. However, the proposed orthogonal series estimator requires an optimal choice of a dimension parameter depending on certain characteristics of f and φ, which are not known in practice. The main issue addressed in our work is the adaptive choice of this dimension parameter using a model selection approach. In a first step, we develop a penalized minimum contrast estimator supposing the degree of ill-posedness of the underlying inverse problem to be known, which amounts to assuming partial knowledge of the error distribution. We show that this data-driven estimator can attain the lower risk bound up to a constant in both sample sizes n and m over a wide range of density classes covering in particular ordinary and super smooth densities. Finally, by randomizing the penalty and the collection of models, we modify the estimator such that it does not require any prior knowledge of the error distribution anymore. Even when dispensing with any hypotheses on φ,this fully data-driven estimator still preserves minimax optimality in almost the same cases as the partially adaptive estimator.


Mathematical Methods of Statistics | 2010

Adaptive estimation in circular functional linear models

Fabienne Comte; Jan Johannes

We consider the problem of estimating the value l(ρ) of a linear functional, where the structural function ρ models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function ρ and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepskis method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function ρ.

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Rudolf Schenk

Université catholique de Louvain

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Maik Schwarz

Université catholique de Louvain

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Nicolas Asin

Université catholique de Louvain

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Sébastien Van Bellegem

Université catholique de Louvain

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Sébastien Van Bellegem

Université catholique de Louvain

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Fabienne Comte

Paris Descartes University

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