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We consider the nonparametric estimation of the value of a quadratic functional evaluated at the density of a strictly positive random variable $X$ based on an iid. sample from an observation $Y$ of $X$ corrupted by an independent multiplicative error $U$. Quadratic functionals of the density covered are the $\mathbb{L}^2$-norm of the density and its derivatives or the survival function. We construct a fully data-driven estimator when the error density is known. The plug-in estimator is based on a density estimation combining the estimation of the Mellin transform of the $Y$ density and a spectral cut-off regularized inversion of the Mellin transform of the error density. The main issue is the data-driven choice of the cut-off parameter using a Goldenshluger-Lepski-method. We discuss conditions under which the fully data-driven estimator attains oracle-rates up to logarithmic deteriorations. We compute convergence rates under classical smoothness assumptions and illustrate them by a simulation study., Comment: 42 page

We consider a multiplicative deconvolution problem, in which the density $f$ or the survival function $S^X$ of a strictly positive random variable $X$ is estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y = X\cdot U$ of $X$. The multiplicative measurement error $U$ is supposed to be independent of $X$. The objective of this work is to construct a fully data-driven estimation procedure when the error density $f^U$ 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. The proposed estimation procedure combines the estimation of the Mellin transformation of the density $f$ and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The derived risk bounds and oracle-type inequalities cover both - the estimation of the density $f$ as well as the survival function $S^X$. The main issue addressed in this work is the data-driven choice of the cut-off parameter using a model selection approach. We discuss conditions under which the fully data-driven estimator can attain the oracle-risk up to a constant without any previous knowledge of the error distribution. We compute convergences rates under classical smoothness assumptions. We illustrate the estimation strategy by a simulation study with different choices of distributions.

We consider the estimation of a density at a fixed point under a local differential privacy constraint, where the observations are anonymised before being available for statistical inference. We propose both a privatised version of a projection density estimator as well as a kernel density estimator and derive their minimax rates under a privacy constraint. There is a twofold deterioration of the minimax rates due to the anonymisation, which we show to be unavoidable by providing lower bounds. In both estimation procedures a tuning parameter has to be chosen. We suggest a variant of the classical Goldenshluger-Lepski method for choosing the bandwidth and the cut-off dimension, respectively, and analyse its performance. It provides adaptive minimax-optimal (up to log-factors) estimators. We discuss in detail how the lower and upper bound depend on the privacy constraints, which in turn is reflected by a modification of the adaptive method.

We study the non-parametric estimation of the value ${\theta}(f )$ of a linear functional evaluated at an unknown density function f with support on $R_+$ based on an i.i.d. sample with multiplicative measurement errors. The proposed estimation procedure combines the estimation of the Mellin transform of the density $f$ and a regularisation of the inverse of the Mellin transform by a spectral cut-off. In order to bound the mean squared error we distinguish several scenarios characterised through different decays of the upcoming Mellin transforms and the smoothnes of the linear functional. In fact, we identify scenarios, where a non-trivial choice of the upcoming tuning parameter is necessary and propose a data-driven choice based on a Goldenshluger-Lepski method. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the estimator., Comment: 25 pages

In a circular deconvolution model we consider the fully data driven density estimation of a circular random variable where the density of the additive independent measurement error is unknown. We have at hand two independent iid samples, one of the contaminated version of the variable of interest, and the other of the additive noise. We show optimality,in an oracle and minimax sense, of a fully data-driven weighted sum of orthogonal series density estimators. Two shapes of random weights are considered, one motivated by a Bayesian approach and the other by a well known model selection method. We derive non-asymptotic upper bounds for the quadratic risk and the maximal quadratic risk over Sobolev-like ellipsoids of the fully data-driven estimator. We compute rates which can be obtained in different configurations for the smoothness of the density of interest and the error density. The rates (strictly) match the optimal oracle or minimax rates for a large variety of cases, and feature otherwise at most a deterioration by a logarithmic factor. We illustrate the performance of the fully data-driven weighted sum of orthogonal series estimators by a simulation study., Comment: 19 pages, 4 figures, 27 appendix pages

We study the non-parametric estimation of an unknown density f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed fully data driven procedure is based on the estimation of the Mellin transform of the density f , a regularisation of the inverse of the Mellin transform by a spectral cut-off and a data-driven model selection in order to deal with the upcoming bias-variance trade-off. We introduce and discuss further Mellin-Sobolev spaces which characterize the regularity of the unknown density f through the decay of its Mellin transform. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the data-driven density estimator and hence its adaptivity., Comment: 22 pages, 2 figures

Given observations from a circular random variable contaminated by an additive measurement error, we consider the problem of minimax optimal goodness-of-fit testing in a non-asymptotic framework. We propose direct and indirect testing procedures using a projection approach. The structure of the optimal tests depends on regularity and ill-posedness parameters of the model, which are unknown in practice. Therefore, adaptive testing strategies that perform optimally over a wide range of regularity and ill-posedness classes simultaneously are investigated. Considering a multiple testing procedure, we obtain adaptive i.e. assumption-free procedures and analyse their performance. Compared with the non-adaptive tests, their radii of testing face a deterioration by a log-factor. We show that for testing of uniformity this loss is unavoidable by providing a lower bound. The results are illustrated considering Sobolev spaces and ordinary or super smooth error densities.

In a circular convolution model, we aim to infer on the density of a circular random variable using observations contaminated by an additive measurement error. We highlight the interplay of the two problems: optimal testing and quadratic functional estimation. Under general regularity assumptions, we determine an upper bound for the minimax risk of estimation for the quadratic functional. The upper bound consists of two terms, one that mimics a classical bias-variance trade-off and a second that causes the typical elbow effect in quadratic functional estimation. Using a minimax optimal estimator of the quadratic functional as a test statistic, we derive an upper bound for the nonasymptotic minimax radius of testing for nonparametric alternatives. Interestingly, the term causing the elbow effect in the estimation case vanishes in the radius of testing. We provide a matching lower bound for the testing problem. By showing that any lower bound for the testing problem also yields a lower bound for the quadratic functional estimation problem, we obtain a lower bound for the risk of estimation. Lastly, we prove a matching lower bound for the term causing the elbow effect in the estimation problem. The results are illustrated considering Sobolev spaces and ordinary or super smooth error densities.

In the inverse Gaussian sequence space model with additional noisy observations of the operator, we derive nonasymptotic minimax radii of testing for ellipsoid-type alternatives simultaneously for both the signal detection problem (testing against zero) and the goodness-of-fit testing problem (testing against a prescribed sequence) without any regularity assumption on the null hypothesis. The radii are the maximum of two terms, each of which only depends on one of the noise levels. Interestingly, the term involving the noise level of the operator explicitly depends on the null hypothesis and vanishes in the signal detection case. The minimax radii are established by first showing a lower bound for arbitrary null hypotheses and noise levels. For the upper bound we consider two testing procedures, a direct test based on estimating the energy in the image space and an indirect test. Under mild assumptions, we prove that the testing radius of the indirect test achieves the lower bound, which shows the minimax optimality of the radius and the test. We highlight the assumptions under which the direct test also performs optimally. Furthermore, we apply a classical Bonferroni method for making both the indirect and the direct test adaptive with respect to the regularity of the alternative. The radii of the adaptive tests are deteriorated by an additional log-factor, which we show to be unavoidable. The results are illustrated considering Sobolev spaces and mildly or severely ill-posed inverse problems.

We study non-parametric estimation of an unknown density with support in R (respectively R+). The proposed estimation procedure is based on the projection on finite dimensional subspaces spanned by the Hermite (respectively the Laguerre) functions. The focus of this paper is to introduce a data-driven aggregation approach in order to deal with the upcoming bias-variance trade-off. Our novel procedure integrates the usual model selection method as a limit case. We show the oracle- and the minimax-optimality of the data-driven aggregated density estimator and hence its adaptivity. We present results of a simulation study which allow to compare the finite sample performance of the data-driven estimators using model selection compared to the new aggregation.

We consider the estimation of a structural function which models a non-parametric relationship between a response and an endogenous regressor given an instrument in presence of dependence in the data generating process. Assuming an independent and identically distributed (iid.) sample it has been shown in Johannes and Schwarz (2010) that a least squares estimator based on dimension reduction and thresholding can attain minimax-optimal rates of convergence up to a constant. As this estimation procedure requires an optimal choice of a dimension parameter with regard amongst others to certain characteristics of the unknown structural function we investigate its fully data-driven choice based on a combination of model selection and Lepski's method inspired by Goldenshluger and Lepski (2011). For the resulting fully data-driven thresholded least squares estimator a non-asymptotic oracle risk bound is derived by considering either an iid. sample or by dismissing the independence assumption. In both cases the derived risk bounds coincide up to a constant assuming sufficiently weak dependence characterised by a fast decay of the mixing coefficients. Employing the risk bounds the minimax optimality up to constant of the estimator is established over a variety of classes of structural functions., Comment: 53 pages

We consider the problem of estimating the slope parameter in functional linear instrumental regression, where in the presence of an instrument W, i.e., an exogenous random function, a scalar response Y is modeled in dependence of an endogenous random function X. Assuming second order stationarity jointly for X and W a nonparametric estimator of the functional slope parameter and its derivatives is proposed based on an n-sample of (Y,X,W). In this paper the minimax optimal rate of convergence of the estimator is derived assuming that the slope parameter belongs to the well-known Sobolev space of periodic functions. We discuss the cases that the cross-covariance operator associated to the random functions X and W is finitely, infinitely or in some general form smoothing., Comment: arXiv admin note: text overlap with arXiv:0901.4266

We consider non-parametric estimation problems in the presence of dependent data, notably non-parametric regression with random design and non-parametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterized by fast decreasing mixing coefficients. 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 function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski (2011). We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients., Comment: 39 pages, 4 figures

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 $\theta^\circ$ that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both $\theta^\circ$ 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 $\theta^\circ$ 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 $\theta^\circ$ 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 $\theta^\circ$ nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

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.

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 $\sigma$. 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 $\sigma$. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.

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.

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., Comment: Published in at http://dx.doi.org/10.1214/12-AOS1050 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider the problem of estimating the value l({\phi}) of a linear functional, where the structural function {\phi} 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 {\phi} 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 {\phi}.

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.

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 $\varphi$ 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 $\varphi$, 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 $\varphi$, 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., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ422 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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.

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.

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.

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., Comment: Revised version for J. of Multivariate Analysis

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.

We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density. We assume that an additional sample $\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\epsilon}$ 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 $f_{\epsilon}$, 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 $f_{\epsilon}$ is ordinary smooth or supersmooth., Comment: Published in at http://dx.doi.org/10.1214/08-AOS652 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)