NAVARRO Fabien

In this paper, we deal with the estimation of an unknown function from a nonparametric regression model with both additive and multiplicative noises. The case of the uniform multiplicative noise is considered. We develop a projection es-timator based on wavelets for this problem. We prove that it attains a fast rate of convergence under the mean integrated square error over Besov spaces. A practical extension to automatically select the truncation parameter of this estimator is discussed. A numerical study illustrates the usefulness of this extension.

This work is motivated by the analysis of accelerometer data. The analysis of such data consists in extracting statistics which characterize the physical activity of a subject (e.g., the mean time spent at different activity levels and the probability of the transition between two levels). Therefore , we introduce a finite mixture model of hidden Markov chain to analyze accelerometer data by considering heterogeneity into the population. This approach does not specify activity levels in advance but estimates them from the data. In addition, it allows for the heterogeneity of the population to be taken into account and defines subpopulations having a homogeneous behavior regarding the physical activity. The main theoretical result is that, under mild assumptions, the probability of misclassifying an observation decreases at an exponential rate with its length. Moreover , we prove the model identifiability and we show how the model can handle missing values. Our proposition is illustrated using real data.

This paper develops a computationally efficient parametric approach to the estimation of general hidden Markov models (HMMs). For non-Gaussian HMMs, the calculation of the Maximum Likelihood Estimator (MLE) involves a high-dimensional integral without an explicit solution that is difficult to calculate with precision. We develop a new alternative method based on the theory of estimating functions and deconvolution strategy. Our procedure requires the same assumptions as the MLE and deconvolution estimators. We provide theoretical guarantees on the performance of the resulting estimator. its consistency and asymptotic normality are established. This leads to building confidence intervals in practice. Monte Carlo experiments are investigated and compared with the MLE. Finally, we illustrate our approach on real data for ex-ante interest rate forecasts.

In this paper, we consider an unknown functional estimation problem in a general nonparametric regression model with the characteristic of having both multiplicative and additive noise. We propose two wavelet estimators, which, to our knowledge, are new in this general context. We prove that they achieve fast convergence rates under the mean integrated square error over Besov spaces. The rates obtained 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 function) supports the theory.

This paper is devoted to adaptive signal denoising in the context of Graph Signal Processing (GSP) using Spectral Graph Wavelet Transform (SGWT). This issue is addressed via a data-driven thresholding process in the transformed domain by optimizing the parameters in the sense of the Mean Square Error (MSE) using the Stein's Unbiased Risk Estimator (SURE). The SGWT considered is built upon a partition of unity making the transform semi-orthogonal so that the optimization can be performed in the transformed domain. However, since the SGWT is over-complete, the SURE needs to be adapted to the context of correlated noise. Two thresholding strategies called coordinatewise and block thresholding process are investigated. For each of them, the SURE is derived for a whole family of elementary thresholding functions among which the soft threshold and the James-Stein threshold. To provide a fully data-driven method, a noise variance estimator derived from the Von Neumann estimator in the Gaussian model is adapted to the graph setting.

In this paper, we deal with the estimation of an unknown function from a nonparametric regression model with both additive and multiplicative noises. The case of the uniform multiplicative noise is considered. We develop a projection es-timator based on wavelets for this problem. We prove that it attains a fast rate of convergence under the mean integrated square error over Besov spaces. A practical extension to automatically select the truncation parameter of this estimator is discussed. A numerical study illustrates the usefulness of this extension.

In this article, a new family of graph wavelets, abbreviated LocLets for Localized graph waveLets, is introduced. These wavelets are localized in the Fourier domain on subsets of the graph Laplacian spectrum. LocLets are built upon the Spectral Graph Wavelet Transform (SGWT) and adapt better to signals that are sparse in the Fourier domain than standard SGWT. In fact, as a refinement of SGWT, LocLets benefits from the Chebyshev's machinery to ensure the LocLets transform remains an efficient and scalable tool for signal processing on large graphs. In addition, LocLets exploits signals sparsity in various ways: compactness, efficiency and ease of use of the transform are improved for sparse signals in the Fourier domain. As typical examples of such sparse signals, there are smooth and highly non-smooth signals. For these latter signals, their mixtures or even a wider class of signals, it is shown in this paper that LocLets provide substantial improvements in standard noise reduction tasks compared to advanced graph-wavelet based methods.

« Hello Dave, you’re looking well today ».

In this note, we introduce new simple approximations for Gaussian type integrals. A key ingredient is the approximation of the function e −x 2 by sum of three simple polynomial-exponential functions. Five special Gaussian type integrals are then considered as applications. Approximation of the Voigt error function is investigated.

Many interesting functional bases, such as piecewise polynomials or wavelets, are examples of localized bases. We investigate the optimality of V-fold cross-validation and a variant called V-fold penalization in the context of the selection of linear models generated by localized bases in a heteroscedastic framework. It appears that while V-fold cross-validation is not asymptotically optimal when V is fixed, the V-fold penalization procedure is optimal. Simulation studies are also presented.

In this note we consider the estimation of the differential entropy of a probability density function. We propose a new adaptive estimator based on a plug-in approach and wavelet methods. We prove that it attains fast rates of convergence under the mean Lp error, p ≥ 1, for a wide class of functions. A key result in our development is a new upper bound for the mean Lp error, p ≥ 1, of a general version of the plug-in estimator. This upper bound may be of independent interest.

We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$-fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1-V^{-1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$-fold cross-validation and $V$-fold penalization have comparable efficiency.

The fact that a slight over-penalization leads to a stabilization of model selection procedures is a phenomenon well known by specialists. Indeed, it has been noticed since the end of the 70's that adding a small positive quantity to classical penalized criteria such as AIC improves in good cases the prediction results, especially for small or moderate sample sizes. The main reason is that over-penalization tends to guard against over-learning. We propose the first general and theoretically sound over-penalization strategy and apply it to the AIC criterion. Very good results are observed by simulation.

This paper deals with the problem of estimating the derivatives of a regression function based on biased data. We develop two different linear wavelet estimators according to the knowledge of the "biased density" of the design. The new estimators are analyzed with respect to their $L_p$ risk with p ≥ 1 over Besov balls. Fast polynomial rates of convergence are obtained.

In this paper we provide a theoretical contribution to the point-wise mean squared error of an adaptive multidimensional term-by-term thresholding wavelet estimator. A general result exhibiting fast rates of convergence under mild assumptions on the model is proved. It can be applied for a wide range of nonparametric models including possible dependent observations. We give applications of this result for the nonpara-metric regression function estimation problem (with random design) and the conditional density estimation problem.

The notion of localized basis is a fruitful concept in the theory of linear approximation, which unites histograms, piecewise polynomials and compactly supported wavelets, among others. We consider the problem of selecting the number of non-zero coefficients in a localized basis development, for the non-parametric estimation of a regression function, with random design and hetroscedastic noise. We then prove by setting up oracle inequalities the asymptotic optimality of the slope heuristic and of a V-fold penalization strategy. We also show that the classical V-fold cross-validation procedure is asymptotically suboptimal in the sense that it finds in...ni the oracle constructed from a fraction of the initial data equal to (V-1)/V. We conclude the paper with a simulatory study on wavelet models, where we note a noticeable difference between the asymptotic results of the theoretical framework and the finite distance practice. Indeed, the V-fold cross-validation and its variant by penalization give in our experiments comparable results while the asymptotic superiority of the penalization is proven.

We investigate the estimation of the $\ell$-fold convolution of the density of an unobserved variable $X$ from $n$ i.i.d. observations of the convolution model $Y=X+\varepsilon$. We first assume that the density of the noise $\varepsilon$ is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of $X$ and $\varepsilon$, we prove that the parametric rate of convergence $1/n$ can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.

This paper attempts to better understand the influence of the smoothness of the copula density in the bi-dimensional estimation density problem. We provide an element of answer by studying the MISE properties of an adaptive estimator based on a plug-in approach and wavelet methods.

We investigate the estimation of the $\ell$-fold convolution of the density of an unobserved variable $X$ from $n$ i.i.d. observations of the convolution model $Y=X+\varepsilon$. We first assume that the density of the noise $\varepsilon$ is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of $X$ and $\varepsilon$, we prove that the parametric rate of convergence $1/n$ can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.

We investigate the estimation of the density-weighted average derivative from biased data. An estimator integrating a plug-in approach and wavelet projections is constructed. We prove that it attains the parametric rate of convergence 1/n under the mean squared error.

The nonparametric estimation of the m-fold convolution power of an unknown function f is considered. We introduce an estimator based on a plug-in approach and a wavelet hard thresholding estimator. We explore its theoretical asymptotic performances via the mean integrated squared error assuming that f has a certain degree of smoothness. Applications and numerical examples are given for the standard density estimation problem and the deconvolution density estimation problem.

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations.

We observe $n$ heteroscedastic stochastic processes $\{Y_v(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t \in [0,1]$, $Y_v(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_v$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_1,\ldots,g_n$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the "BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions.

The problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the $\rho$-mixing case and the $\alpha$-mixing case. More precisely, denoting $n$ the number of observations, in the independent case, we prove that it attains $1/n$ under the mean squared error, in the $\rho$-mixing case, $1/\sqrt{n}$ under the mean absolute error, and, in the $\alpha$-mixing case, $\sqrt{\ln n /n}$ under the mean absolute error. A short simulation study illustrates the theory.

The present paper is concerned with the problem of estimating the convolution of densities. We propose an adaptive estimator based on kernel methods, Fourier analysis and the Lepski method. We study its $\mathbb{L}_2$-risk properties. Fast and new rates of convergence are determined for a wide class of unknown functions. Numerical illustrations, on both simulated and real data, are provided to assess the performances of our estimator.

We investigate the estimation of the integral of the square of a multidimensional unknown function $f$ under mild assumptions on the model allowing dependence on the observations. We develop an adaptive estimator based on a plug-in approach and wavelet projections. Taking the mean absolute error and assuming that $f$ has a certain degree of smoothness, we prove that our estimator attains a sharp rate of convergence. Applications are given for the biased density model, the nonparametric regression model and a GARCH-type model under some mixing dependence conditions ($\alpha$-mixing or $\beta$-mixing). A simulation study considering nonparametric regression models with dependent observations illustrates the usefulness of the proposed estimator.

In this paper, we propose a data-driven block thresholding procedure for wavelet- based non-blind deconvolution. The approach consists in appropriately writing the problem in the wavelet domain and then selecting both the block size and threshold parameter at each resolution level by minimizing Stein's unbiased risk estimate. The resulting algorithm is simple to implement and fast. Numerical illustrations are provided to assess the performance of the estimator.

This thesis presents new statistical procedures in a non-parametric framework and studies both their theoretical and empirical properties. Our work focuses on two different topics which have in common the indirect observation of the unknown functional parameter. The first part is devoted to a block thresholding estimation procedure for the white Gaussian noise model. We focus on the adaptive estimation of a signal f and its derivatives from n fuzzy and noisy versions of this signal and prove that our estimator reaches a (quasi-)optimal convergence speed on a large class of Besov balls. Then, we propose an adaptive estimation procedure allowing to select the parameters of the estimator in an optimal way by minimizing an unbiased risk estimator. In the second part, we place ourselves in the framework of the density model in which the unknown function undergoes a given transformation before being observed. We construct and study an adaptive estimator based on a plug-in approach and a hard block wavelet thresholding. Finally, we study the problem of estimating the m-order convolution of a density from observations i. I. D. drawn from the underlying law. We propose an adaptive estimator based on kernels, Fourier analysis and Lepski's method. We study its quadratic risk and new and fast speeds are obtained, for a large class of unknown functions. Each estimation method studied is numerically analyzed by simulations, both on simulated data and on real data.

This thesis presents new statistical procedures in a non-parametric framework and studies both their theoretical and empirical properties. Our work focuses on two different topics which have in common the indirect observation of the unknown functional parameter. The first part is devoted to a block thresholding estimation procedure for the white Gaussian noise model. We focus on the adaptive estimation of a signal f and its derivatives from n fuzzy and noisy versions of this signal and prove that our estimator reaches a (quasi-)optimal convergence speed on a large class of Besov balls. Then, we propose an adaptive estimation procedure allowing to select the parameters of the estimator in an optimal way by minimizing an unbiased risk estimator. In the second part, we place ourselves in the framework of the density model in which the unknown function undergoes a given transformation before being observed. We construct and study an adaptive estimator based on a plug-in approach and a hard block wavelet thresholding. Finally, we study the problem of estimating the m-order convolution of a density from i.i.d. observations drawn from the underlying law. We propose an adaptive estimator based on kernels, Fourier analysis and Lepski's method. We study its quadratic risk and new and fast speeds are obtained, for a large class of unknown functions. Each estimation method studied is numerically analyzed by simulations, both on simulated data and on real data.