Adaptive estimation in the single-index model via oracle approach
Oleg Lepski, Nora Serdyukova
备注：
28 pages. Lemma 1 is common to "Adaptive estimation under single-index constraint in a regression model." as well as Theorem 4 coincides with Theorem 5 in that paper
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. We suggest a novel procedure that adapts simultaneously to the unknown index and smoothness of the link function. For the proposed procedure, we prove a "local" oracle inequality (described by the pointwise seminorm), which is then used to obtain the upper bound on the maximal risk of the adaptive estimator under assumption that the link function belongs to a scale of Hölder classes. The lower bound on the minimax risk shows that in the case of estimating at a given point the constructed estimator is optimally rate adaptive over the considered range of classes. For the same procedure we also establish a "global" oracle inequality (under the $ L_r $ norm, $r< \infty $) and examine its performance over the Nikol'skii classes. This study shows that the proposed method can be applied to estimating functions of inhomogeneous smoothness, that is whose smoothness may vary from point to point.
An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed.
In one-dimensional density estimation on i.i.d. observations we suggest an adaptive cross-validation technique for the selection of a kernel estimator. This estimator is both asymptotic MISE-efficient with respect to the monotone oracle, and sharp minimax-adaptive over the whole scale of Sobolev spaces with smoothness index greater than 1/2. The proof of the central concentration inequality avoids "chaining" and relies on an additive decomposition of the empirical processes involved.
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 Lp-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm inde…
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 main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].
The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift--finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich--Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estima…
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 link function. For the proposed procedure, we present a "local" oracle inequality (described by the pointwise seminorm), which is then used to obtain the upper bound on the maximal risk under regularity assumption on the link function. The lower boun…
Zou [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and the initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle l…
This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi - martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein -- Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.
This paper considers the problem of estimating a periodic function in a continuous time regression model with a general square integrable semimartingale noise. A model selection adaptive procedure is proposed. Sharp non-asymptotic oracle inequalities have been derived.
An adaptive nonparametric estimation procedure is constructed for heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (oracle inequality) is obtained
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 collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary $\rL_p$--losses. These results are applied for adaptive estimation in the additive multi--index model.
We consider the problem of model selection and estimation in situations where the number of parameters diverges with the sample size. When the dimension is high, an ideal method should have the oracle property [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and [Ann. Statist. 32 (2004) 928--961] which ensures the optimal large sample performance. Furthermore, the high-dimensionality often induces the collinearity problem, which should be properly handled by the ideal method. Many existing variable selection methods fail to achieve both goals simultaneously. In this paper, we propose the adapti…
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 conce…
This paper presents an adaptive version of the Hill estimator based on Lespki's model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand's concentration inequality for smooth functions of independent exponentially distributed random variables combined with three to…
We consider non parametric estimation problem for stochastic tomography regression model, i.e. we consider the estimation problem of function of multivariate variables (image) observed through its Radon transformation calculated with the random errors. For this problem we develop a new adaptive model selection method. By making use the Galtchouk and Pergamenshchikov approach we construct the model selection procedure for which we show a sharp non asymptotic oracle inequality for the both usual and robust quadratic risks, i.e. we show that the proposed procedure is optimal in the oracle inequal…
We study the problem of nonparametric estimation under $\bL_p$-loss, $p\in [1,\infty)$, in the framework of the convolution structure density model on $\bR^d$. 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 family of "kernel-type" estimators is proposed. For the selected estimator, we prove an $\bL_p$-norm oracle inequality and several of its consequences. In Part II the problem of adaptive minimax estimation under $\bL_p$--loss over the sca…
We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a "linear oracle" -- an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2$-norm of the oracle filter is small enough, such oracle can be "mimicked" by an efficiently computable adaptive estimate of the same structure with an observation-driven filter. The filter in question was obtained as a solution to the optimization problem in which the $\ell_\inf…
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 results contain, as a particular case, the complete solution of the bandwidth selection problem in multivariate density model. Usefulness of the developed approach is illustrated by appli- cation to adaptive estimation over anisotropic Nikolskii clas…
We consider the problem of estimating the tail index $α$ of a distribution satisfying a $(α, β)$ second-order Pareto-type condition, where βis the second-order coefficient. When $β$ is available, it was previously proved that $α$ can be estimated with the oracle rate $n^{-β/(2β+1)}$. On the contrary, when $β$ is not available, estimating $α$ with the oracle rate is challenging; so additional assumptions that imply the estimability of $β$ are usually made. In this paper, we propose an adaptive estimator of $α$, and show that this estimator attains the rate $(n/\log\log n)^{-β/(2β+1)}$ without a…
A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the traditional zero expectation assumption. We propose a penalization method based on the quantile loss function with appropriately chosen penalty function making inference on possibly sparse high-dimensional quantile vector. We apply a local approach to address the optimality by comparing procedures to the oracle sparsity structure. We establish that the proposed pro…
Let $(\bX, Y)$ be a random pair taking values in $\mathbb R^p \times \mathbb R$. In the so-called single-index model, one has $Y=f^{\star}(θ^{\star T}\bX)+\bW$, where $f^{\star}$ is an unknown univariate measurable function, $θ^{\star}$ is an unknown vector in $\mathbb R^d$, and $W$ denotes a random noise satisfying $\mathbb E[\bW|\bX]=0$. The single-index model is known to offer a flexible way to model a variety of high-dimensional real-world phenomena. However, despite its relative simplicity, this dimension reduction scheme is faced with severe complications as soon as the underlying dimens…
The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the model misspecification error. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of aff…
University of California, San Diego（加州大学圣地亚哥分校）；Duke University（杜克大学）；Institute for Mathematics and its Applications（数学及其应用研究所）
This paper describes a novel theoretical characterization of the performance of non-local means (NLM) for noise removal. NLM has proven effective in a variety of empirical studies, but little is understood fundamentally about how it performs relative to classical methods based on wavelets or how various parameters (e.g., patch size) should be chosen. For cartoon images and images which may contain thin features and regular textures, the error decay rates of NLM are derived and compared with those of linear filtering, oracle estimators, variable-bandwidth kernel methods, Yaroslavsky's filter an…
We focus on the nonparametric density estimation problem with directional data. We propose a new rule for bandwidth selection for kernel density estimation. Our procedure is automatic, fully data-driven and adaptive to the smoothness degree of the density. We obtain an oracle inequality and optimal rates of convergence for the L2 error. Our theoretical results are illustrated with simulations.
We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_p$ loss. An application to the problem of estimating a signal or its $r^{th}$ derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Si…
The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (``global'' oracle inequality). Both the results are applied to the problems of pointwise and global adaptive estima…
We develop a new model selection method for the adaptive robust efficient nonparametric signal estimation observed with impulse noise which is defined by the general non Gaussian Lévy processes. On the basis of the developed method, we construct the estimation procedures which are analyzed in two settings: in non asymptotic and asymptotic ones. For the first time for such models we show non asymptotic sharp oracle inequalities for the quadratic and for the robust risks, i.e. we show that the constructed procedures are optimal in the sharp oracle inequalities sense. Next, by making use of the o…
We consider the problem of pointwise estimation of multi-dimensional signals $s$, from noisy observations $(y_τ)$ on the regular grid $\bZd$. Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal $s$ is "well-filtered", i.e. there exists an adapted time-invariant linear filter $q^*_T$ with the coefficients which vanish outside the "cube" $\{0,..., T\}^d$ which recovers $s_0…
In this paper, we study the Adaptive Elastic Net method for the Cox model. We prove the grouping effect and oracle property of its estimators. Finally, we show these two properties by an empirical analysis and a numerical simulation, respectively.
The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,γ}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\tilde{f}_{n,γ}$. Then, minimax properties of $\tilde{f}_{n,γ}$ are established. We first prove that the rate…
We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model selection procedure is proposed. Under general moment conditions on the noise distribution a sharp non-asymptotic oracle inequality for the robust risks is obtained and the robust efficiency is shown. It turns out that for semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
We provide in this paper a fully adaptive penalized procedure to select a covariance among a collection of models observing i.i.d replications of the process at fixed observation points. For this we generalize previous results of Bigot and al. and propose to use a data driven penalty to obtain an oracle inequality for the estimator. We prove that this method is an extension to the matricial regression model of the work by Baraud.
We show that under a linearity condition on the distribution of the predictors, the coefficient in single-index regression can be estimated with the same efficiency as in the case when the link function is known. Thus, the linearity condition seems to substitute for knowing the exact conditional distribution of the response given the linear combinations of the predictors.
This paper deals with estimation with functional covariates. More precisely, we aim at estimating the regression function $m$ of a continuous outcome $Y$ against a standard Wiener coprocess $W$. Following Cadre and Truquet (2015) and Cadre, Klutchnikoff, and Massiot (2017) the Wiener-Itô decomposition of $m(W)$ is used to construct a family of estimators. The minimax rate of convergence over specific smoothness classes is obtained. A data-driven selection procedure is defined following the ideas developed by Goldenshluger and Lepski (2011). An oracle-type inequality is obtained which leads to …
In this paper, we consider the robust adaptive non parametric estimation problem for the drift coefficient in diffusion processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Sharp oracle inequalities for the robust risk have been obtained.
A two-class mixture model, where the density of one of the components is known, is considered. We address the issue of the nonparametric adaptive estimation of the unknown probability density of the second component. We propose a randomly weighted kernel estimator with a fully data-driven bandwidth selection method, in the spirit of the Goldenshluger and Lepski method. An oracle-type inequality for the pointwise quadratic risk is derived as well as convergence rates over Holder smoothness classes. The theoretical results are illustrated by numerical simulations.
We consider the problem of statistical learning for the intensity of a counting process with covariates. In this context, we introduce an empirical risk, and prove risk bounds for the corresponding empirical risk minimizers. Then, we give an oracle inequality for the popular algorithm of aggregation with exponential weights. This provides a way of constructing estimators that are adaptive to the smoothness and to the structure of the intensity. We prove that these estimators are adaptive over anisotropic Besov balls. The probabilistic tools are maximal inequalities using the generic chaining m…
In this article we consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises observed in discrete time moments. An adaptive model selection procedure is proposed. A sharp non-asymptotic oracle inequality for the robust risks is obtained. We obtain sufficient conditions on the frequency observations under which the robust efficiency is shown. It turns out that for the semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_1$-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_1$-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the …
We propose an algorithm to estimate the common density $s$ of a stationary process $X_1,...,X_n$. We suppose that the process is either $β$ or $τ$-mixing. We provide a model selection procedure based on a generalization of Mallows' $C_p$ and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d framework.
University of Virginia（弗吉尼亚大学）；University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
This paper proposes a Lasso-type estimator for a high-dimensional sparse parameter identified by a single index conditional moment restriction (CMR). In addition to this parameter, the moment function can also depend on a nuisance function, such as the propensity score or the conditional choice probability, which we estimate by modern machine learning tools. We first adjust the moment function so that the gradient of the future loss function is insensitive (formally, Neyman-orthogonal) with respect to the first-stage regularization bias, preserving the single index property. We then take the l…
Flexible estimation of heterogeneous treatment effects lies at the heart of many statistical challenges, such as personalized medicine and optimal resource allocation. In this paper, we develop a general class of two-step algorithms for heterogeneous treatment effect estimation in observational studies. We first estimate marginal effects and treatment propensities in order to form an objective function that isolates the causal component of the signal. Then, we optimize this data-adaptive objective function. Our approach has several advantages over existing methods. From a practical perspective…
