In this paper, we study the following model of hidden Markov chain: Y_i=X_i+ɛ_i, i=1,…,n+1 with (X_i) a real-valued stationary Markov chain and (ɛ_i)_1≤i≤n+1 a noise having a known distribution and independent of the sequence (X_i). We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The L²-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators.

Parameter estimation in a class of heteroscedastic time series models is investigated. The existence of conditional least-squares and conditional likelihood estimators is proved. Their consistency and their asymptotic normality are established. Kernel estimators of the noise’s density and its derivatives are defined and shown to be uniformly consistent. A simulation experiment conducted shows that the estimators perform well for large sample size.

We consider nonparametric Bayesian estimation of a probability density p based on a random sample of size n from this density using a hierarchical prior. The prior consists, for instance, of prior weights on the regularity of the unknown density combined with priors that are appropriate given that the density has this regularity. More generally, the hierarchy consists of prior weights on an abstract model index and a prior on a density model for each model index. We present a general theorem on the rate of contraction of the resulting posterior distribution as n→∞, which gives conditions under which the rate of contraction is the one attached to the model that best approximates the true density of the observations. This shows that, for instance, the posterior distribution can adapt to the smoothness of the underlying density. We also study the posterior distribution of the model index, and find that under the same conditions the posterior distribution gives negligible weight to models that are bigger than the optimal one, and thus selects the optimal model or smaller models that also approximate the true density well. We apply these result to log spline density models, where we show that the prior weights on the regularity index interact with the priors on the models, making the exact rates depend in a complicated way on the priors, but also that the rate is fairly robust to specification of the prior weights.

We derive the l_∞ convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two different assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small.

When applying the support vector machine (SVM) to high-dimensional classification problems, we often impose a sparse structure in the SVM to eliminate the influences of the irrelevant predictors. The lasso and other variable selection techniques have been successfully used in the SVM to perform automatic variable selection. In some problems, there is a natural hierarchical structure among the variables. Thus, in order to have an interpretable SVM classifier, it is important to respect the heredity principle when enforcing the sparsity in the SVM. Many variable selection methods, however, do not respect the heredity principle. In this paper we enforce both sparsity and the heredity principle in the SVM by using the so-called structured variable selection (SVS) framework originally proposed in [20]. We minimize the empirical hinge loss under a set of linear inequality constraints and a lasso-type penalty. The solution always obeys the desired heredity principle and enjoys sparsity. The new SVM classifier can be efficiently fitted, because the optimization problem is a linear program. Another contribution of this work is to present a nonparametric extension of the SVS framework, and we propose nonparametric heredity SVMs. Simulated and real data are used to illustrate the merits of the proposed method.

A theorem of Donsker asserts that the empirical process converges in distribution to the Brownian bridge. The aim of this paper is to provide a new and simple proof of this fact.

The discovering of low-dimensional manifolds in high-dimensional data is one of the main goals in manifold learning. We propose a new approach to identify the effective dimension (intrinsic dimension) of low-dimensional manifolds. The scale space viewpoint is the key to our approach enabling us to meet the challenge of noisy data. Our approach finds the effective dimensionality of the data over all scale without any prior knowledge. It has better performance compared with other methods especially in the presence of relatively large noise and is computationally efficient.

The Support Vector Machine (SVM) is a popular classification paradigm in machine learning and has achieved great success in real applications. However, the standard SVM can not select variables automatically and therefore its solution typically utilizes all the input variables without discrimination. This makes it difficult to identify important predictor variables, which is often one of the primary goals in data analysis. In this paper, we propose two novel types of regularization in the context of the multicategory SVM (MSVM) for simultaneous classification and variable selection. The MSVM generally requires estimation of multiple discriminating functions and applies the argmax rule for prediction. For each individual variable, we propose to characterize its importance by the supnorm of its coefficient vector associated with different functions, and then minimize the MSVM hinge loss function subject to a penalty on the sum of supnorms. To further improve the supnorm penalty, we propose the adaptive regularization, which allows different weights imposed on different variables according to their relative importance. Both types of regularization automate variable selection in the process of building classifiers, and lead to sparse multi-classifiers with enhanced interpretability and improved accuracy, especially for high dimensional low sample size data. One big advantage of the supnorm penalty is its easy implementation via standard linear programming. Several simulated examples and one real gene data analysis demonstrate the outstanding performance of the adaptive supnorm penalty in various data settings.

Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying clustering structures. Hence removing noise variables via variable selection is necessary. For simultaneous variable selection and parameter estimation, existing penalized likelihood approaches in model-based clustering analysis all assume a common diagonal covariance matrix across clusters, which however may not hold in practice. To analyze high-dimensional data, particularly those with relatively low sample sizes, this article introduces a novel approach that shrinks the variances together with means, in a more general situation with cluster-specific (diagonal) covariance matrices. Furthermore, selection of grouped variables via inclusion or exclusion of a group of variables altogether is permitted by a specific form of penalty, which facilitates incorporating subject-matter knowledge, such as gene functions in clustering microarray samples for disease subtype discovery. For implementation, EM algorithms are derived for parameter estimation, in which the M-steps clearly demonstrate the effects of shrinkage and thresholding. Numerical examples, including an application to acute leukemia subtype discovery with microarray gene expression data, are provided to demonstrate the utility and advantage of the proposed method.

In several application fields like daily pluviometry data modelling, or motion analysis from image sequences, observations contain two components of different nature. A first part is made with discrete values accounting for some symbolic information and a second part records a continuous (real-valued) measurement. We call such type of observations “mixed-state observations".

This paper introduces spatial models suited for the analysis of these kinds of data. We consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family. Specific examples with exponential distributions are detailed, and we present some experimental results for modelling motion measurements from video sequences.

This paper is devoted to the estimation of a vector θ parametrizing an energy function of a Gibbs point process, via the maximum pseudolikelihood method. Strong consistency and asymptotic normality results of this estimator depending on a single realization are presented. In the framework of exponential family models, sufficient conditions are expressed in terms of the local energy function and are verified on a wide variety of examples.

Let X₁,…,X_n be i.i.d. observations, where X_i=Y_i+σZ_i and Y_i and Z_i are independent. Assume that unobservable Y’s are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Z_i have the standard normal distribution and let σ>0. Based on a sample X₁,…,X_n, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leibler property of the Dirichlet mixture prior has been shown for some special kernels like the normal density or Bernstein polynomial, under appropriate conditions. In this paper, we obtain easily verifiable sufficient conditions, under which a prior obtained by mixing a general kernel possesses the Kullback-Leibler property. We study a wide variety of kernel used in practice, including the normal, t, histogram, gamma, Weibull densities and so on, and show that the Kullback-Leibler property holds if some easily verifiable conditions are satisfied at the true density. This gives a catalog of conditions required for the Kullback-Leibler property, which can be readily used in applications.

The purpose of this paper is to join two different threads of the recent literature on random fields on the sphere, namely the statistical analysis of higher order angular power spectra on one hand, and the construction of second-generation wavelets on the sphere on the other. To this aim, we introduce the needlets bispectrum and we derive a number of convergence results. Here, the limit theory is developed in the high resolution sense. The leading motivation of these results is the need for statistical procedures for searching non-Gaussianity in Cosmic Microwave Background radiation.

Multivariate statistics are often available as well as necessary in hypothesis tests. We study how to use such statistics to control not only false discovery rate (FDR) but also positive FDR (pFDR) with good power. We show that FDR can be controlled through nested regions of multivariate p-values of test statistics. If the distributions of the test statistics are known, then the regions can be constructed explicitly to achieve FDR control with maximum power among procedures satisfying certain conditions. On the other hand, our focus is where the distributions are only partially known. Under certain conditions, a type of nested regions are proposed and shown to attain (p)FDR control with asymptotically maximum power as the pFDR control level approaches its attainable limit. The procedure based on the nested regions is compared with those based on other nested regions that are easier to construct as well as those based on more straightforward combinations of the test statistics.

This paper deals with the problem of function estimation. Using the white noise model setting, we provide a method to construct a new wavelet procedure based on thresholding rules which takes advantage of the dyadic structure of the wavelet decomposition. We prove that this new procedure performs very well since, on the one hand, it is adaptive and near-minimax over a large class of Besov spaces and, on the other hand, the maximal functional space (maxiset) where this procedure attains a given rate of convergence is very large. More than this, by studying the shape of its maxiset, we prove that the new procedure outperforms the hard thresholding procedure.

A wealth of epidemiological data suggests an association between mortality/morbidity from pulmonary and cardiovascular adverse events and air pollution, but uncertainty remains as to the extent implied by those associations although the abundance of the data. In this paper we describe an SSA (Singular Spectrum Analysis) based approach in order to decompose the time-series of particulate matter concentration into a set of exposure variables, each one representing a different timescale. We implement our methodology to investigate both acute and long-term effects of PM₁₀ exposure on morbidity from respiratory causes within the urban area of Bari, Italy.

In this paper we introduce an influence measure based on second order expansion of the RV and GCD measures for the comparison between unperturbed and perturbed eigenvectors of a symmetric matrix estimator. Example estimators are considered to highlight how this measure compliments recent influence analysis. Importantly, we also show how a sample based version of this measure can be used to accurately and efficiently detect influential observations in practice.

Let (X,Y) be a random variable consisting of an observed feature vector $X\in \mathcal{X}$ and an unobserved class label Y∈{1,2,…,L} with unknown joint distribution. In addition, let $\mathcal{D}$ be a training data set consisting of n completely observed independent copies of (X,Y). Usual classification procedures provide point predictors (classifiers) $\widehat{Y}(X,\mathcal{D})$ of Y or estimate the conditional distribution of Y given X. In order to quantify the certainty of classifying X we propose to construct for each θ=1,2,…,L a p-value $\pi_{\theta}(X,\mathcal{D})$ for the null hypothesis that Y=θ, treating Y temporarily as a fixed parameter. In other words, the point predictor $\widehat{Y}(X,\mathcal{D})$ is replaced with a prediction region for Y with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.

The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data.

It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function [10]. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.

We investigate in this paper the estimation of Gaussian graphs by model selection from a non-asymptotic point of view. We start from an n-sample of a Gaussian law ℙ_C in ℝ^p and focus on the disadvantageous case where n is smaller than p. To estimate the graph of conditional dependences of ℙ_C, we introduce a collection of candidate graphs and then select one of them by minimizing a penalized empirical risk. Our main result assesses the performance of the procedure in a non-asymptotic setting. We pay special attention to the maximal degree D of the graphs that we can handle, which turns to be roughly n/(2logp).

Inference based on the penalized density ratio model is proposed and studied. The model under consideration is specified by assuming that the log–likelihood function of two unknown densities is of some parametric form. The model has been extended to cover multiple samples problems while its theoretical properties have been investigated using large sample theory. A main application of the density ratio model is testing whether two, or more, distributions are equal. We extend these results by arguing that the penalized maximum empirical likelihood estimator has less mean square error than that of the ordinary maximum likelihood estimator, especially for small samples. In fact, penalization resolves any existence problems of estimators and a modified Wald type test statistic can be employed for testing equality of the two distributions. A limited simulation study supports further the theory.

Consider a random vector (X,Y) and let m(x)=E(Y|X=x). We are interested in testing $H_{0}:m\in {\cal M}_{\Theta,{\cal G}}=\{\gamma(\cdot,\theta,g):\theta \in \Theta,g\in {\cal G}\}$ for some known function γ, some compact set $\Theta \subset I\!\!R^{p}$ and some function set ${\cal G}$ of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis.

To test this null hypothesis, we make use of the so-called marked empirical process introduced by [4] and studied by [16] for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.

We establish estimation and model selection consistency, prediction and estimation bounds and persistence for the group-lasso estimator and model selector proposed by Yuan and Lin (2006) for least squares problems when the covariates have a natural grouping structure. We consider the case of a fixed-dimensional parameter space with increasing sample size and the double asymptotic scenario where the model complexity changes with the sample size.

In Markov chain Monte Carlo (MCMC) sampling considerable thought goes into constructing random transitions. But those transitions are almost always driven by a simulated IID sequence. Recently it has been shown that replacing an IID sequence by a weakly completely uniformly distributed (WCUD) sequence leads to consistent estimation in finite state spaces. Unfortunately, few WCUD sequences are known. This paper gives general methods for proving that a sequence is WCUD, shows that some specific sequences are WCUD, and shows that certain operations on WCUD sequences yield new WCUD sequences. A numerical example on a 42 dimensional continuous Gibbs sampler found that some WCUD inputs sequences produced variance reductions ranging from tens to hundreds for posterior means of the parameters, compared to IID inputs.

We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence without prior knowledge of the regularity of the function to be estimated. We provide for such estimators oracle inequalities and optimal rates of convergence. This penalized approach is applied to Tikhonov regularization and to regularization by projection.

Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alternative approach to FPCA using penalized rank one approximation to the data matrix. Our contributions are four-fold: (1) by considering invariance under scale transformation of the measurements, the new formulation sheds light on how regularization should be performed for FPCA and suggests an efficient power algorithm for computation; (2) it naturally incorporates spline smoothing of discretized functional data; (3) the connection with smoothing splines also facilitates construction of cross-validation or generalized cross-validation criteria for smoothing parameter selection that allows efficient computation; (4) different smoothing parameters are permitted for different FPCs. The methodology is illustrated with a real data example and a simulation.

In this paper we investigate the performance of periodogram based estimators of the spectral density matrix of possibly high-dimensional time series. We suggest and study shrinkage as a remedy against numerical instabilities due to deteriorating condition numbers of (kernel) smoothed periodogram matrices. Moreover, shrinking the empirical eigenvalues in the frequency domain towards one another also improves at the same time the Mean Squared Error (MSE) of these widely used nonparametric spectral estimators. Compared to some existing time domain approaches, restricted to i.i.d. data, in the frequency domain it is necessary to take the size of the smoothing span as “effective or local sample size” into account. While Böhm & von Sachs (2007) proposes a multiple of the identity matrix as optimal shrinkage target in the absence of knowledge about the multidimensional structure of the data, here we consider “structural” shrinkage. We assume that the spectral structure of the data is induced by underlying factors. However, in contrast to actual factor modelling suffering from the need to choose the number of factors, we suggest a model-free approach. Our final estimator is the asymptotically MSE-optimal linear combination of the smoothed periodogram and the parametric estimator based on an underfitting (and hence deliberately misspecified) factor model. We complete our theoretical considerations by some extensive simulation studies. In the situation of data generated from a higher-order factor model, we compare all four types of involved estimators (including the one of Böhm & von Sachs (2007)).

We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range dependence, as opposite to the standard case, when no noise is present. In particular, if the dependence is moderate the bandwidth, the rates of mean-square convergence and, additionally, central limit theorem are the same as in the i.i.d. case. If the dependence is strong enough, then the bandwidth choice is influenced by the strength of dependence, which is different when compared to the non-noisy case. Also, central limit theorem are influenced by the strength of dependence. On the other hand, if the density is supersmooth, then long range dependence has no effect at all on the optimal bandwidth choice.

We consider the classification problem on the cube [0,1]^d when the Bayes rule is known to belong to some new functions classes. These classes are made of prediction rules satisfying some conditions regarding their coefficients when developed over the (overcomplete) basis of indicator functions of dyadic cubes of [0,1]^d. The main concern of the paper is on the thorough analysis of the approximation term, which is in general bypassed in the classification literature. An adaptive classifier is designed to achieve the minimax rate of convergence (up to a logarithmic factor) over these functions classes. Lower bounds on the convergence rate over these classes are established when the underlying marginal of the design is comparable to the Lebesgue measure. Connections with some existing models for classification (RKHS and “boundary fragements”) are established.

Case-deleted analysis is a popular method for evaluating the influence of a subset of cases on inference. The use of Monte Carlo estimation strategies in complicated Bayesian settings leads naturally to the use of importance sampling techniques to assess the divergence between full-data and case-deleted posteriors and to provide estimates under the case-deleted posteriors. However, the dependability of the importance sampling estimators depends critically on the variability of the case-deleted weights. We provide theoretical results concerning the assessment of the dependability of case-deleted importance sampling estimators in several Bayesian models. In particular, these results allow us to establish whether or not the estimators satisfy a central limit theorem. Because the conditions we derive are of a simple analytical nature, the assessment of the dependability of the estimators can be verified routinely before estimation is performed. We illustrate the use of the results in several examples.

We analyse the properties of the Principal Fitted Components (PFC) algorithm proposed by Cook. We derive theoretical properties of the resulting estimators, including sufficient conditions under which they are $\sqrt{n}$-consistent, and explain some of the simulation results given in Cook’s paper. We use techniques from random matrix theory and perturbation theory. We argue that, under Cook’s model at least, the PFC algorithm should outperform the Principal Components algorithm.

Based on the methods provided in Caeiro and Gomes (2002) and Fraga Alves (2001), a new class of location invariant Hill-type estimators is derived in this paper. Its asymptotic distributional representation and asymptotic normality are presented, and the optimal choice of sample fraction by Mean Squared Error is also discussed for some special cases. Finally comparison studies are provided for some familiar models by Monte Carlo simulations.

We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate n^{−γ/(2γ+1)} of convergence if the true density of the observations belongs to the Hölder space C^γ[0,1]. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model.

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. To achieve this, we establish an exponential inequality for the Nadaraya-Watson variance function estimator.

We discuss various implementation issues concerning our wavelet estimator, and demonstrate its good practical performance. We also show how it leads to a new wavelet-domain data-driven variance-stabilising transform. Our estimator can be applied to a variety of problems, including the estimation of volatilities, spectral densities and Poisson intensities, as well as to a range of problems in which the distribution of the noise is unknown.

We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density f and some partially known noise density g. In this work, g is assumed exponentially smooth with stable law having unknown self-similarity index s. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities f, with smoothness parameter β. In this context, we first provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional ∫f² and goodness-of-fit test of the hypothesis H₀:f=f₀, where the alternative H₁ is expressed with respect to $\mathbb{L}_{2}$-norm (i.e. has the form $\psi_{n}^{-2}\|f-f_{0}\|_{2}^{2}\ge \mathcal{C}$). These procedures are adaptive with respect to both s and β and attain the rates which are known optimal for known values of s and β. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of s is illustrated on synthetic data.

We present a sequential Monte Carlo sampler algorithm for the Bayesian analysis of generalised linear mixed models (GLMMs). These models support a variety of interesting regression-type analyses, but performing inference is often extremely difficult, even when using the Bayesian approach combined with Markov chain Monte Carlo (MCMC). The Sequential Monte Carlo sampler (SMC) is a new and general method for producing samples from posterior distributions. In this article we demonstrate use of the SMC method for performing inference for GLMMs. We demonstrate the effectiveness of the method on both simulated and real data, and find that sequential Monte Carlo is a competitive alternative to the available MCMC techniques.

A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphasize the smooth attributes of the model. A formula is $Q_n^{OPT}=n∑_{j=1}^{p_n}j^{−1/2}‖𝒀 _{n,j}‖^2$. Down-weighting by $j^{−1/2}$ is selected to achieve adaptive optimality among tests based on tapering with respect to its “rates of testing,” an asymptotic framework for measuring a test’s retention of power in high dimensions under smoothness constraints. Existing tests based on truncation or thresholding are known to have superior asymptotic power in comparison with any test based on tapering; however, it is shown here that high-order effects can be substantial, and that a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the sort of alternatives that would typically be of concern in functional data analysis applications. The proposed test is developed for use in practice, and demonstrated in an example application.

We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call “self-consistency condition”, concerns the algorithm itself, and the second, called “dependency control condition” is related to the dependency assumptions on the p-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, p-value reweighting, new family of step-up procedures under unspecified p-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.

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. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator.

Given an IID sample from a positive distribution, we provide a method for constructing rigorous finite sample lower confidence bounds for the expectation of the distribution. The method is based on constructing rigorous confidence regions for the cdf of the distribution. We provide some analysis of the asymptotical behavior of the rigorous LCBs. We apply the method to obtain an LCB for a particular, controversial, empirical data set, where the validity of standard methods has been called into question.

For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the 100(1−α)% Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\frac{3\alpha}{2}$ and $1-\frac{3\alpha}{2}+\frac{\alpha^{2}}{1+\alpha}$; with the lower bound $1-\frac{3\alpha}{2}$ improving (for α≤1/3) on the previously established ([9]; [8]) lower bound $\frac{1-\alpha}{1+\alpha}$. Several illustrative examples are given.

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed h>0 and z∈ℝ^d: $$G_{n}(K,h,z):=\sum_{i=1}^{n}K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)-\mathbb{E}\Bigl(K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)\Big),$$ where the Z_i are i.i.d. on ℝ^d. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\mathcal{G}$ and on the law of Z₁, we establish a uniform functional limit law for the collections of processes $\bigl\{G_{n}(\cdot,h_{n},z),\;z\in H,\;h\in [h_{n},\mathfrak{h}_{n}]\big\}$, where H⊂ℝ^d is a compact set with nonempty interior and where h_n and $\mathfrak{h}_{n}$ satisfy the Csörgő-Révész-Stute conditions.

We investigate the performance of a family of multiple comparison procedures for strong control of the False Discovery Rate ($\mathsf{FDR}$). The $\mathsf{FDR}$ is the expected False Discovery Proportion ($\mathsf{FDP}$), that is, the expected fraction of false rejections among all rejected hypotheses. A number of refinements to the original Benjamini-Hochberg procedure [1] have been proposed, to increase power by estimating the proportion of true null hypotheses, either implicitly, leading to one-stage adaptive procedures [4, 7] or explicitly, leading to two-stage adaptive (or plug-in) procedures [2, 21].

We use a variant of the stochastic process approach proposed by Genovese and Wasserman [11] to study the fluctuations of the $\mathsf{FDP}$ achieved with each of these procedures around its expectation, for independent tested hypotheses.

We introduce a framework for the derivation of generic Central Limit Theorems for the $\mathsf{FDP}$ of these procedures, characterizing the associated regularity conditions, and comparing the asymptotic power of the various procedures. We interpret recently proposed one-stage adaptive procedures [4, 7] as fixed points in the iteration of well known two-stage adaptive procedures [2, 21].

We deal with the estimation of the regime number in a linear Gaussian autoregressive process with a Markov regime (AR-MR). The problem of estimating the number of regimes in this type of series is that of determining the number of states in the hidden Markov chain controlling the process. We propose a method based on penalized maximum likelihood estimation and establish its strong consistency (almost sure) without assuming previous bounds on the number of states.

We propose a general family of algorithms for regression estimation with quadratic loss, on the basis of geometrical considerations. These algorithms are able to select relevant functions into a large dictionary. We prove that a lot of methods that have already been studied for this task (LASSO, Dantzig selector, Iterative Feature Selection, among others) belong to our family, and exhibit another particular member of this family that we call Correlation Selector in this paper. Using general properties of our family of algorithm we prove oracle inequalities for IFS, for the LASSO and for the Correlation Selector, and compare numerical performances of these estimators on a toy example.

This paper investigates correct variable selection in finite samples via ℓ₁ and ℓ₁+ℓ₂ type penalization schemes. The asymptotic consistency of variable selection immediately follows from this analysis. We focus on logistic and linear regression models. The following questions are central to our paper: given a level of confidence 1−δ, under which assumptions on the design matrix, for which strength of the signal and for what values of the tuning parameters can we identify the true model at the given level of confidence? Formally, if Î is an estimate of the true variable set I^*, we study conditions under which ℙ(Î=I^*)≥1−δ, for a given sample size n, number of parameters M and confidence 1−δ. We show that in identifiable models, both methods can recover coefficients of size $\frac{1}{\sqrt{n}}$, up to small multiplicative constants and logarithmic factors in M and $\frac{1}{\delta}$. The advantage of the ℓ₁+ℓ₂ penalization over the ℓ₁ is minor for the variable selection problem, for the models we consider here. Whereas the former estimates are unique, and become more stable for highly correlated data matrices as one increases the tuning parameter of the ℓ₂ part, too large an increase in this parameter value may preclude variable selection.

In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\mathbb{L}_{2}$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets.

We consider a l₁-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension d of the input variable X is very large (sometimes depending on the number of observations). Estimation of a β-regular regression function f cannot be faster than the slow rate n^{−2β/(2β+d)}. Hopefully, in some situations, f depends only on a few numbers of the coordinates of X. In this paper, we construct two procedures. The first one selects, with high probability, these coordinates. Then, using this subset selection method, we run a local polynomial estimator (on the set of interesting coordinates) to estimate the regression function at the rate n^{−2β/(2β+d*)}, where d^*, the “real” dimension of the problem (exact number of variables whom f depends on), has replaced the dimension d of the design. To achieve this result, we used a l₁ penalization method in this non-parametric setup.

Ordinary differential equations (ODE’s) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE’s from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.

Relative surprise regions are shown to minimize, among Bayesian credible regions, the prior probability of covering a false value from the prior. Such regions are also shown to be unbiased in the sense that the prior probability of covering a false value is bounded above by the prior probability of covering the true value. Relative surprise regions are shown to maximize both the Bayes factor in favor of the region containing the true value and the relative belief ratio, among all credible regions with the same posterior content. Relative surprise regions emerge naturally when we consider equivalence classes of credible regions generated via reparameterizations.

Upper bounds for rates of convergence of posterior distributions associated to Gaussian process priors are obtained by van der Vaart and van Zanten in [14] and expressed in terms of a concentration function involving the Reproducing Kernel Hilbert Space of the Gaussian prior. Here lower-bound counterparts are obtained. As a corollary, we obtain the precise rate of convergence of posteriors for Gaussian priors in various settings. Additionally, we extend the upper-bound results of [14] about Riemann-Liouville priors to a continuous family of parameters.

Two concepts of symmetry for the distributions of positive random variables Y are log-symmetry (symmetry of the distribution of logY) and R-symmetry [7]. In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric.

In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, F, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of F are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of F are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of F have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of F have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, F̂, of F is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.

Let Y be a Gaussian vector of ℝⁿ of mean s and diagonal covariance matrix Γ. Our aim is to estimate both s and the entries σ_i=Γ_i,i, for i=1,…,n, on the basis of the observation of two independent copies of Y. Our approach is free of any prior assumption on s but requires that we know some upper bound γ on the ratio max _iσ_i/min _iσ_i. For example, the choice γ=1 corresponds to the homoscedastic case where the components of Y are assumed to have common (unknown) variance. In the opposite, the choice γ>1 corresponds to the heteroscedastic case where the variances of the components of Y are allowed to vary within some range. Our estimation strategy is based on model selection. We consider a family {S_m×Σ_m, m∈ℳ} of parameter sets where S_m and Σ_m are linear spaces. To each m∈ℳ, we associate a pair of estimators (ŝ_m,σ̂_m) of (s,σ) with values in S_m×Σ_m. Then we design a model selection procedure in view of selecting some m̂ among ℳ in such a way that the Kullback risk of (ŝ_m̂,σ̂_m̂) is as close as possible to the minimum of the Kullback risks among the family of estimators {(ŝ_m,σ̂_m), m∈ℳ}. Then we derive uniform rates of convergence for the estimator (ŝ_m̂,σ̂_m̂) over Hölderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.

Recently, Giraitis et al. (2003, [10]) proposed the V/S statistic for testing long memory in random sequences. We generalize this statistic to the setting of random fields. The null hypothesis is concerned with short memory random fields while the alternative contains a very large family of long memory random fields. Contrary to most of the previous works dealing with long-range dependence, no assumption is made about the isotropy of the strong dependence. Some simulations are presented in order to assess the power of the test according to the kind of long memory in presence.