arXiv subjects

Jan Johannes

Publications and source records attributed to Jan Johannes.

32 records · Page 2Linked to original sources

arXiv2015

Adaptive Bayesian estimation in indirect Gaussian sequence space models

In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value $θ^\circ$ that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both $θ^\circ$ and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each $θ^\circ$ separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive. This means that, given a parameter $θ^\circ$ or a class of parameters, respectively, the posterior distribution contracts at the oracle rate or at the minimax rate over the class. Notably, the hierarchical prior does not depend neither on $θ^\circ$ nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

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arXiv2013

Adaptive circular deconvolution by model selection under unknown error distribution

We consider a circular deconvolution problem, in which the density $f$ of a circular random variable $X$ must be estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement error is supposed to be independent of $X$. The objective of this work was to construct a fully data-driven estimation procedure when the error density $φ$ is unknown. We assume that in addition to the i.i.d. sample from $Y$, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. We first develop a minimax theory in terms of both sample sizes. We propose an orthogonal series estimator attaining the minimax rates but requiring 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 this 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 assuming that the error density is known. We show that this partially adaptive estimator can attain the lower risk bound up to a constant in both sample sizes $n$ and $m$. Finally, by randomizing the penalty and the collection of models, we modify the estimator such that it no longer requires any previous knowledge of the error distribution. 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. We illustrate our results by computing minimal rates under classical smoothness assumptions.

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arXiv2013

Iterative Estimation of Solutions to Noisy Nonlinear Operator Equations in Nonparametric Instrumental Regression

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.

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arXiv2013

Adaptive functional linear regression

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 minimax-rates 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.

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arXiv2012

Partially adaptive nonparametric instrumental regression

We consider the problem of estimating the structural function in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the structural function belongs to some ellipsoids which are in a certain sense linked to the conditional expectation operator of Z given W. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the unknown structural function and the conditional expectation operator of Z given W, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter using a model selection approach under the restriction that the conditional expectation operator of Z given W is smoothing in a certain sense. In this situation we develop a penalized minimum contrast estimator with randomized penalty and collection of models. We show that this data-driven estimator can attain the lower risk bound up to a constant over a wide range of smoothness classes for the structural function.

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arXiv2012

Adaptive Gaussian inverse regression with partially unknown operator

This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\varepsilon$ and $σ$. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of $f$ and $A$. This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method. We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for $f$ and $A$ and noise levels $\varepsilon$ and $σ$. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.

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arXiv2011

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of random functions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure 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 these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski's method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as the minimizer of a stochastic penalized contrast function imitating Lepski's method among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers point-wise estimation as well as the estimation of local averages of the slope parameter.

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arXiv2011

Adaptive estimation of functionals in nonparametric instrumental regression

We consider the problem of estimating the value l(ϕ) of a linear functional, where the structural function ϕ 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 ϕ 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 Lepski's 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 ϕ.

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arXiv2009

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of 1-periodic, second order stationary random functions X1,...,Xn. We consider an orthogonal series estimator of the slope function, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. Wepropose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L2-risk. This means that we provide adaptive estimators of both the slope function and its derivatives.

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arXiv2009

Deconvolution with unknown error distribution

We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_ε$, where $f_ε$ is an unknown density. We assume that an additional sample $ε_1,...,ε_m$ from $f_ε$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_ε$ 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 $f_ε$, where it is assumed that $f_X$ 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 $f_X$ belongs to a Sobolev space $H_{\mathbh p}$ and $f_ε$ is ordinary smooth or supersmooth.

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arXiv2009

On rate optimal local estimation in nonparametric instrumental regression

We consider the problem of estimating the value of a linear functional in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the structural function and the representer of the linear functional belong to some ellipsoids which are in a certain sense linked to the conditional expectation operator of Z given W. We illustrate these results by considering classical smoothness assumptions.

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arXiv2009

Nonparametric estimation in functional linear models with second order stationary regressors

We consider the problem of estimating the slope parameter in functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of second order stationary random functions X1,...,Xn. An orthogonal series estimator of the functional slope parameter with additional thresholding in the Fourier domain is proposed and its performance is measured with respect to a wide range of weighted risks covering as examples the mean squared prediction error and the mean integrated squared error for derivative estimation. In this paper the minimax optimal rate of convergence of the estimator is derived over a large class of different regularity spaces for the slope parameter and of different link conditions for the covariance operator. These general results are illustrated by the particular example of the well-known Sobolev space of periodic functions as regularity space for the slope parameter and the case of finitely or infinitely smoothing covariance operator.

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arXiv2008

Thresholding Projection Estimators in Functional Linear Models

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 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 to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove these estimators are minimax and rates of convergence are given for some particular cases.

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arXiv2007

Nonparametric estimation for dependent data with an application to panel time series

In this paper we consider nonparametric estimation for dependent data, where the observations do not necessarily come from a linear process. We study density estimation and also discuss associated problems in nonparametric regression using the 2-mixing dependence measure. We compare the results under 2-mixing with those derived under the assumption that the process is linear. In the context of panel time series where one observes data from several individuals, it is often too strong to assume the joint linearity of processes. Instead the methods developed in this paper enable us to quantify the dependence through 2-mixing which allows for nonlinearity. We propose an estimator of the panel mean function and obtain its rate of convergence. We show that under certain conditions the rate of convergence can be improved by allowing the number of individuals in the panel to increase with time.

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