In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.
Nonparametric density estimators are studied for $d$-dimensional, strongly spatial mixing data which is defined on a general $N$-dimensional lattice structure. We consider linear and nonlinear hard thresholded wavelet estimators which are derived from a $d$-dimensional multiresolution analysis. We give sufficient criteria for the consistency of these estimators and derive rates of convergence in $L^{p'}$ for $p'\in [1,\infty)$. For this reason, we study density functions which are elements of a $d$-dimensional Besov space $B^s_{p,q}(\mathbb{R}^d)$. We also verify the analytic correctness of ou…
We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz technique, consists of estimating the variance function via a Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure which uses the estimated variance function and local means of the data to set the thresholds at a suitable level. We demonstrate the mean-square near-optimality of our wavelet estimator over the usual range of Besov classes. T…
We consider the problem of constructing robust nonparametric confidence intervals and tests of hypothesis for the median when the data distribution is unknown and the data may contain a small fraction of contamination. We propose a modification of the sign test (and its associated confidence interval) which attains the nominal significance level (probability coverage) for any distribution in the contamination neighborhood of a continuous distribution. We also define some measures of robustness and efficiency under contamination for confidence intervals and tests. These measures are computed fo…
In this paper, a nonparametric estimator is proposed for estimating the L1-median for multivariate conditional distribution when the covariates take values in an infinite dimensional space. The multivariate case is more appropriate to predict the components of a vector of random variables simultaneously rather than predicting each of them separately. While estimating the conditional L1-median function using the well-known Nadarya-Waston estimator, we establish the strong consistency of this estimator as well as the asymptotic normality. We also present some simulations and provide how to built…
We consider the nonparametric estimation problem of time-dependent multivariate functions observed in a presence of additive cylindrical Gaussian white noise of a small intensity. We derive minimax lower bounds for the $L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero, when the underlying unknown response function is assumed to belong to a ball of appropriately constructed inhomogeneous time-dependent multivariate functions, motivated by practical applications. Furthermore, we propose both non-adaptive linear and adaptive non-linear wavelet estimators that are asym…
The problem of constructing confidence regions for the median in the nonparametric measurement error model (NMEM) is considered. This problem arises in many settings, including inference about the median lifetime of a complex system arising in engineering, reliability, biomedical, and public health settings. Current methods of constructing CRs are discussed, including the T-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably the two default methods in applied work when a confidence interval about the center of a distribution is desired. Optimal equivariant CRs are deve…
A theory of superefficiency and adaptation is developed under flexible performance measures which give a multiresolution view of risk and bridge the gap between pointwise and global estimation. This theory provides a useful benchmark for the evaluation of spatially adaptive estimators and shows that the possible degree of superefficiency for minimax rate optimal estimators critically depends on the size of the neighborhood over which the risk is measured. Wavelet procedures are given which adapt rate optimally for given shrinking neighborhoods including the extreme cases of mean squared error …
Consider nonparametric function estimation under $L^p$-loss. The minimax rate for estimation of the regression function over a Hölder ball with smoothness index $β$ is $n^{-β/(2β+1)}$ if $1\leq p<\infty$ and $(n/\log n)^{-β/(2β+1)}$ if $p=\infty.$ There are many known procedures that either attain this rate for $p=\infty$ but are suboptimal by a $\log n$ factor in the case $p<\infty$ or the other way around. In this article, we construct an estimator that simultaneously achieves the optimal rates under $L^p$-risk for all $1\leq p\leq \infty$ without prior knowledge of $β.$ In contrast to class…
We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on functional kernel nonparametric regression estimation techniques where observations are segments of the observed process considered as curves. These curves are assumed to lie within a space of possibly inhomogeneous functions, and the discretized times series dataset consists of a relatively small, compared to the number of segments, number of measurements made at regular times. We thus consider only the case where an asymptoticall…
We study nonparametric estimation of the diffusion coefficient from discrete data, when the observations are blurred by additional noise. Such issues have been developed over the last 10 years in several application fields and in particular in high frequency financial data modelling, however mainly from a parametric and semiparametric point of view. This paper addresses the nonparametric estimation of the path of the (possibly stochastic) diffusion coefficient in a relatively general setting. By developing pre-averaging techniques combined with wavelet thresholding, we construct adaptive estim…
Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed. The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: …
For signals belonging to balls in smoothness classes and noise with enough moments, the asymptotic behavior of the minimax quadratic risk among soft-threshold estimates is investigated. In turn, these results, combined with a median filtering method, lead to asymptotics for denoising heavy tails via wavelet thresholding. Some further comparisons of wavelet thresholding and of kernel estimators are also briefly discussed.
We consider density estimation for Besov spaces when each sample is quantized to only a limited number of bits. We provide a noninteractive adaptive estimator that exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minimax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which …
We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Grübel (2003) and Duval (2…
This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the $L^{2}$-risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.
We study the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over $[0,T]$. We consider the microscopic regime when the sampling rate $Δ=Δ_T\rightarrow0$ as $T\rightarrow\infty$. We propose an adaptive wavelet threshold density estimator and study its performance for the $L_p$ loss, $p\geq 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates $Δ_T$ that vanish with $T$ at arbitrary polynomial rates. More precicely, our estimator attains minimax rates of convergence pro…
We present a general principle for estimating a regression function nonparametrically, allowing for a wide variety of data filtering, for example, repeated left truncation and right censoring. Both the mean and the median regression cases are considered. The method works by first estimating the conditional hazard function or conditional survivor function and then integrating. We also investigate improved methods that take account of model structure such as independent errors and show that such methods can improve performance when the model structure is true. We establish the pointwise asymptot…
We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, $f\in\mathcal{B}^s_{π,r}$, and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $α,0<α\leq1$; heteroscedasticity is modeled with a design dependent function $σ$. We prescribe a tuning paradigm, under which warped wavelet estimation achieves partial or full adaptivity results with the rates that are show…
In this paper, we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise.We propose two new wavelet estimators in this general context. We prove that they achieve fast convergence rates under the mean integrated square error over Besov spaces. The obtained rates have the particularity of being established under weak conditions on the model. A numerical study in a context comparable to stochastic frontier estimation (with the difference that the boundary is not necessarily a production funct…
Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence resu…
We construct nonparametric confidence sets for regression functions using wavelets that are uniform over Besov balls. We consider both thresholding and modulation estimators for the wavelet coefficients. The confidence set is obtained by showing that a pivot process, constructed from the loss function, converges uniformly to a mean zero Gaussian process. Inverting this pivot yields a confidence set for the wavelet coefficients, and from this we obtain confidence sets on functionals of the regression curve.
In this paper, we introduce a robust nonparametric density estimator combining the popular Kernel Density Estimation method and the Median-of-Means principle (MoM-KDE). This estimator is shown to achieve robustness to any kind of anomalous data, even in the case of adversarial contamination. In particular, while previous works only prove consistency results under known contamination model, this work provides finite-sample high-probability error-bounds without a priori knowledge on the outliers. Finally, when compared with other robust kernel estimators, we show that MoM-KDE achieves competitiv…
Astrophysicists are interested in recovering the 3D gas emissivity of a galaxy cluster from a 2D image taken by a telescope. A blurring phenomenon and presence of point sources make this inverse problem even harder to solve. The current state-of-the-art technique is two step: first identify the location of potential point sources, then mask these locations and deproject the data. We instead model the data as a Poisson generalized linear model (involving blurring, Abel and wavelets operators) regularized by two lasso penalties to induce sparse wavelet representation and sparse point sources. Th…
In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the syste…
From a wavelet analysis, one derives a nonparametrical estimator for the spectral density of a Gaussian process with stationary increments. First, the idealistic case of a continuous time path of the process is considered. A punctual Central Limit Theorem (CLT) and an estimation of the Mean Integrate Square Error (MISE) are established. Next, to fit the applications, one considers the case where one observes a path at random times. One built a second estimator obtained by replacing the wavelet coefficients by their discretizations. A second CLT and the corresponding estimation of the MISE are …
In this paper, we study robust estimators of the memory parameter d of a (possibly) non stationary Gaussian time series with generalized spectral density f. This generalized spectral density is characterized by the memory parameter d and by a function f* which specifies the short-range dependence structure of the process. Our setting is semi-parametric since both f* and d are unknown and d is the only parameter of interest. The memory parameter d is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two estimators of d that we c…
Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks…
This paper develops intrinsic wavelet denoising methods for surfaces of Hermitian positive definite matrices, with in mind the application to nonparametric estimation of the time-varying spectral matrix of a multivariate locally stationary time series. First, we construct intrinsic average-interpolating wavelet transforms acting directly on surfaces of Hermitian positive definite matrices in a curved Riemannian manifold with respect to an affine-invariant metric. Second, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces o…
This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an $l_1$-penalty based wavelet estimator of the nonparametric component and Huber's M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of b…
In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our…
In this paper we provide the asymptotic theory of the general of $ϕ$-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived as well as symmetrized estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particulariz…
We present a novel approach for nonparametric regression using wavelet basis functions. Our proposal, $\texttt{waveMesh}$, can be applied to non-equispaced data with sample size not necessarily a power of 2. We develop an efficient proximal gradient descent algorithm for computing the estimator and establish adaptive minimax convergence rates. The main appeal of our approach is that it naturally extends to additive and sparse additive models for a potentially large number of covariates. We prove minimax optimal convergence rates under a weak compatibility condition for sparse additive models. …
We consider nonparametric estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the singular values of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for $P$ and ${p}$, discarding most of the estimated singular triplets. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establish…
In this paper we consider a class of nonparametric estimators of a distribution function F, with compact support, based on the theory of IFSs. The estimator of F is tought as the fixed point of a contractive operator T defined in terms of a vector of parameters p and a family of affine maps W which can be both depend of the sample (X_1, X_2, ...., X_n). Given W, the problem consists in finding a vector p such that the fixed point of T is ``sufficiently near'' to F. It turns out that this is a quadratic constrained optimization problem that we propose to solve by penalization techniques. If F h…
This paper is about two related decision theoretic problems, nonparametric two-sample testing and independence testing. There is a belief that two recently proposed solutions, based on kernels and distances between pairs of points, behave well in high-dimensional settings. We identify different sources of misconception that give rise to the above belief. Specifically, we differentiate the hardness of estimation of test statistics from the hardness of testing whether these statistics are zero or not, and explicitly discuss a notion of "fair" alternative hypotheses for these problems as dimensio…
We study minimax convergence rates of nonparametric density estimation in the Huber contamination model, in which a proportion of the data comes from an unknown outlier distribution. We provide the first results for this problem under a large family of losses, called Besov integral probability metrics (IPMs), that includes $\mathcal{L}^p$, Wasserstein, Kolmogorov-Smirnov, and other common distances between probability distributions. Specifically, under a range of smoothness assumptions on the population and outlier distributions, we show that a re-scaled thresholding wavelet series estimator a…
The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of a…
A nonparametric procedure for robust regression estimation and for quantile regression is proposed which is completely data-driven and adapts locally to the regularity of the regression function. This is achieved by considering in each point M-estimators over different local neighbourhoods and by a local model selection procedure based on sequential testing. Non-asymptotic risk bounds are obtained, which yield rate-optimality for large sample asymptotics under weak conditions. Simulations for different univariate median regression models show good finite sample properties, also in comparison t…
This paper addresses the problem of regularity properties of functions represented as an expansion in a wavelet basis with random coefficients in terms of finiteness of their Besov norm with probability 1. Such representations are used to specify a prior measure in Bayesian nonparametric wavelet regression. Investigating regularity of such functions is an important problem since the support of the posterior measure does not include functions that are not in the support of the prior measure, and therefore determines the functions that are possible to estimate using specified Bayesian model. We …
We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the tra…
We study the performances of an adaptive procedure based on a convex combination, with data-driven weights, of term-by-term thresholded wavelet estimators. For the bounded regression model, with random uniform design, and the nonparametric density model, we show that the resulting estimator is optimal in the minimax sense over all Besov balls under the $L^2$ risk, without any logarithm factor.
We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators.
We study empirical Bayes estimation of the effect sizes of $N$ units from $K$ noisy observations on each unit. We show that it is possible to achieve near-Bayes optimal mean squared error, without any assumptions or knowledge about the effect size distribution or the noise. The noise distribution can be heteroskedastic and vary arbitrarily from unit to unit. Our proposal, which we call Aurora, leverages the replication inherent in the $K$ observations per unit and recasts the effect size estimation problem as a general regression problem. Aurora with linear regression provably matches the perf…
Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function and all its mixed partial derivatives. We show that their posterior distributions contract to the truth optimally and adaptively under supremum-norm loss. The master theorem through tests with exponential errors used in Bayesian nonparametrics was not adequate to deal with this problem, and we deve…
We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a log…
In the multidimensional setting, we consider the errors-in-variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimator based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic H{ö}lder classes. Our theoretical results are illustrated by some simulations.
In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between …
The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estim…
This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet b…
A simple Bayesian approach to nonparametric regression is described using fuzzy sets and membership functions. Membership functions are interpreted as likelihood functions for the unknown regression function, so that with the help of a reference prior they can be transformed to prior density functions. The unknown regression function is decomposed into wavelets and a hierarchical Bayesian approach is employed for making inferences on the resulting wavelet coefficients.
We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.
Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relat…
We investigate asymptotic properties of least-absolute-deviation or median quantile estimates of the location and scale functions in nonparametric regression models with dependent data from multiple subjects. Under a general dependence structure that allows for longitudinal data and some spatially correlated data, we establish uniform Bahadur representations for the proposed median quantile estimates. The obtained Bahadur representations provide deep insights into the asymptotic behavior of the estimates. Our main theoretical development is based on studying the modulus of continuity of kernel…
