Oleg Lepski

Aix-Marseille Université | AMU · Département de mathématiques

In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate--o...

The paper deals with the problem of nonparametric estimating the $L_p$--norm, $p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from independent observations. The unknown density %to be estimated is assumed to belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax approach, and derive lower bounds on the minimax r...

This paper deals with non-parametric density estimation on R 2 from i.i.d observations. It is assumed that after unknown rotation of the coordinate system the coordinates of the observations are independent random variables whose densities belong to a Hölder class with unknown parameters. The minimax and adaptive minimax theories for this structura...

The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in Rd. Their common probability distribution function p can be written as p = (1-α)f + α[f ∗ g], w...

This paper continues the research started in Lepski and Willer (2016). In the framework of the convolution structure density model on R d , we address the problem of adaptive minimax estimation with Lp–loss over the scale of anisotropic Nikol'skii classes. We fully characterize the behavior of the minimax risk for different relationships between re...

We study the problem of nonparametric estimation under Lp-loss, p ∈ [1, ∞), in the framework of the convolution structure density model on R d. This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. In Part I the original pointwise selection rule from a fami...

In this paper we are interested in finding upper functions for a collection of random variables { ξ ⃗ h p , ⃗ h ∈ H } , 1 ≤ p < ∞. Here ξ ⃗ h (x), x ∈ (−b, b) d , d ≥ 1 is a kernel-type gaussian random field and ∥ · ∥p stands for Lp-norm on (−b, b) d. The set H consists of d-variate vector-functions defined on (−b, b) d and taking values in some co...

In the framework of an abstract statistical model we discuss how to use the solution of one estimation problem ({\it Problem A}) in order to construct an estimator in another, completely different, {\it Problem B}. As a solution of {\it Problem A} we understand a data-driven selection from a given family of estimators $\mathbf{A}(\mH)=\big\{\wideha...

Cours of lectures given at High School of Economics in November 2014

We address the problem of adaptive minimax estimation in white gaussian noise model under $L_p$--loss, $1\leq p\leq\infty,$ on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called...

In this paper we are interested in finding upper functions for a collection of random variables $\big\{\big\|\xi_{\vec{h}}\big\|_p, \vec{h}\in\mathrm{H}\big\}, 1\leq p<\infty$. Here $\xi_{\vec{h}}(x), x\in(-b,b)^d, d\geq 1$ is a kernel-type gaussian random field and $\|\cdot\|_p$ stands for $L_p$-norm on $(-b,b)^d$. The set $\mathrm{H}$ consists of...

In this part of the paper we apply the results obtained in Lepski (2013) to the variety of problems related to empirical processes.

The problem of adaptive multivariate function estimation under single-index assumption is studied in the framework of the regression model with random design. We consider the case when both the link function and index vector are unknown. We propose a novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothnes...

In the framework of nonparametric multivariate function estimation we are interested in structural adaptation. We assume that the function to be estimated possesses the single-index structure where neither the link function nor the index vector is known. We propose a novel procedure that adapts simultaneously to the unknown index and smoothness of...

This paper deals with the density estimation on ℝ d under sup-norm loss. We provide a fully data-driven estimation procedure and establish for it a so-called sup-norm oracle inequality. The proposed estimator allows us to take into account not only approximation properties of the underlying density, but eventual independence structure as well. Our...

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ θ ), θ ∈ Θ}. Here {χ θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ℝ+ such that supθ∈Θ{Ψ(χ θ )...

In the framework of an abstract statistical model we propose a procedure for selecting an estimator from a given family of linear estimators. We derive an upper bound on the risk of the selected estimator and demonstrate how this result can be used in order to develop minimax and adaptive minimax estimators in specific nonparametric estimation prob...

—In this part of the paper we apply the results obtained in Lepski (2013) to the variety of problems related to empirical processes.

In the general statistical experiment model we propose a procedure for selecting an estimator from a given family of linear estimators. We derive an upper bound on the risk of the selected estimator and demontrate how this result can be used in order to construct minimax and adaptive minimax estimators in specic nonparametric estimation problems.

The paper deals with the density estimation on Rd under sup- norm loss. We provide with fully data-driven estimation procedure and establish for it so called sup-norm oracle inequality. The pro- posed estimator allows to take into account not only approximation properties of the underlying density but eventual independence struc- ture as well. Our...

We address the problem of adaptive minimax density estimation on $\bR^d$ with $\bL_p$--loss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are...

The main objective of this paper is to look from the unique point of view at some phenomena arising in different areas of probability theory and mathematical statistics. We will try to understand what is common between classical probabilistic results, such as the law of iterated logarithm for example, and well-known problem in adaptive estimation c...

In the framework of nonparametric multivariate function estimation we are interested in structural adaptation. We assume that the function to be estimated has the “single-index” structure where neither the link function nor the index vector is known. This article suggests a novel procedure that adapts simultaneously to the unknown index and the smo...

We address the problem of density estimation with $\mathbb{L}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The...

We study the problem of nonparametric estimation of a multivariate function $g:\mathbb {R}^d\to\mathbb{R}$ that can be represented as a composition of two unknown smooth functions $f:\mathbb{R}\to\mathbb{R}$ and $G:\mathbb{R}^d\to \mathbb{R}$. We suppose that $f$ and $G$ belong to known smoothness classes of functions, with smoothness $\gamma$ and...

We present a nonparametric test for determining when observed data are consistent with the hypothesis of cost minimizing behavior. The model allows testing in the presence of noisy data. While this problem can be solved in general nonparametric framework, the large distance that the alternatives must be separated from the null hypothesis for the te...

In this paper, we develop a general machinery for finding explicit uniform probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the ${\mathbb{L}}_s$-norms of empirical and regression-type processes. Usefulness of the obtained results is illustrated...

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collect...

In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for differen...

A continuous change-point problem is studied in which N independent diffusion processes X j are observed. Each process X j is associated with a “channel”, each has an unknown piecewise constant drift and the unit diffusion coefficient. All the channels are connected only by a common change-point of drift. As the result, a change-point problem is...

In this paper, we consider the problem of the minimax hypothesis testing in the multivariate white gaussian noise model. We want to test the hypothesis about the absence of the signal against the alternative belonging to the set of smooth composite functions separated away from zero in sup-norm. We propose the test procedure and show that it is opt...

In dimension one, it has long been observed that the minimax rates of conver- gences in the scale of Besov spaces present essentially two regimes (and a boundary): dense and the sparse zones. In this paper, we consider the problem of denoising a function depending of a multidimensional variable (for instance an image), with anisotropic constraints...

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collect...

In dimension one, it has long been observed that the minimax rates of convergences in the scale of Besov spaces present essentially two regimes (and a boundary): dense and the sparse zones. In this paper, we consider the problem of denoising a function depending on a multidimensional variable (for instance, an image), with anisotropic constraints o...

The new approach, allowed to take into account some additional information, coming from datas, is proposed. The main idea is to obtain from datas some information about structure of the model in order to improve accuracy of estimation. It seems to be important, since standard nonparametric accuracy of estimation is usually very low. To improve one...

We consider the problem of solving linear operator equations from noisy data under the assumption that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the sp...

The minimax properties of a test verifying a symmetry of an unknown regression function f from n independent observations are studied. The underlying design is assumed to be random and independent of the noise in observations. The function f belongs to a ball in a Hölder space of regularity β. The null hypothesis accepts that f is symmetric. We tes...

In the context of minimax theory, we propose a new kind of risk, normalized by a random variable, measurable with respect to the data. We present a notion of optimality and a method to construct optimal procedures accordingly. We apply this general setup to the problem of selecting significant variables in Gaussian white noise. In particular, we sh...

In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of the Lepski's method of adaptation, and roughly consists i...

We consider the problem of recovering smooth functions from noisy data, using the sup-norm as the quality criterion. Starting with a natural projection estimator, we show a data-driven procedure to be adaptive asymptotically minimax. 1 Introduction In this paper we study the problem of recovering an unknown function f(t); t 2 [0; 1]; from noisy dat...

Adaptive pointwise estimation of smooth functions f(x) in R is studied in the white Gaussian noise model of a given intensity " ! 0. It is assumed that the Fourier transform of f belongs to a large class of rapidly vanishing functions but is otherwise unknown. Optimal adaptation in higher dimensions presents several challenges. First, the number of...

Stochastic Volatility (SV) models are widely used in financial applications. To decide whether standard parametric restrictions are justified for a given data set, a statistical test is required. In this paper, we develop such test of a linear hypothesis versus a general composite nonparametric alternative using the state space representation of th...

The aim of this paper is to synthetically analyse the performances of thresholding and wavelet estimation methods. In this connection, it is useful to describe the maximal sets where these methods attain a special rate of convergence. We relate these “maxisets” to other problems naturally arising in the context of non parametric estimation, as appr...

We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separat...

be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L r norms ||f|| r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L 2 ) functional or estimating a singular f...

A new approach, allowing one to take into account some additional information coming from the data, is proposed. The main idea is to obtain from the data some information about the structure of the model in order to improve the accuracy of estimation. This can be important, since the standard nonparametric accuracy of estimation is usually very low...

Stochastic Volatility (SV) models are widely used in financial applications. To decide whether standard parametric restrictions are justified for a given dataset, a statistical test is required. In this paper, we develop such a test based on the linear state space representation. We provide a simulation study and apply the test to the HFDF96 data s...

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. ¶ First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and...

A new variable bandwidth selector for kernel estimation is proposed. The application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over Besov classes. This implies that the procedure adapts to spatially inhomogeneous smoothness. In particular, the estimates share optimality properties with wavelet es...

The problem of nonparametric estimation of functions of inhomogeneous smoothness is considered. The aim of the paper is to propose estimation procedures which possess the property of local adaptation in the following sense. A function is recovered from noisy data within each (possibly small) interval with accuracy corresponding to the actual smooth...

This paper deals with testing of nonparametric hypotheses when the model of observation is unknown function [ sigma (.)] plus a Gaussian White Noise with a small diffusion [ epsilon > 0 ]. It is required to distinguish the simple hypothesis H_0 : [ sigma(.) ] = 0 against the composite alternative H_[ epsilon] : [ sigma(.) ][ is an element of ][ sum...

Stochastic Volatility (SV) models are widely used in financial applications. To decide whether standard parametric restrictions are justified for a given dataset, a statistical test is required. In this paper, we develop such a test based on the linear state space representation. We provide a simulation study and apply the test to the HFDF96 data s...

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H 0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H 1 : f∈Λɛ where Λɛ is a set of functions on R 1 of the form Λɛ = {f : f∈?, ϕ(f) ≥ Cψɛ}. Here ? is a Hölder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the val...

Let a function f be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of f , more precisely, L r -norm kfk r of f . Existing in the literature results on estimation of functionals deal mostly with two extreme cases: estimation of a smooth (diierentiable in L 2) functional or...