The last few years have witnessed important new developments inthe theory and practice of pattern classification. We intend tosurvey some of the main new ideas that have led to theserecent results.

The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) isa powerful alternative to EM when the E-step is intractable. Convergence ofSAEM toward a maximum of the observed likelihood is established whenthe unobserved data are simulated at each iteration under the conditionaldistribution. We show that this very restrictive assumption can be weakened. Indeed, the results of Benveniste et al. for stochastic approximationwith Markovian perturbations are used to establish the convergenceof SAEM when it is coupled with a Markov chain Monte-Carloprocedure. This result is very useful for many practical applications. Applications to the convolution model and the change-points model are presented to illustrate the proposed method.

Many mathematical models involve input parameters, which are not precisely known. Globalsensitivity analysis aims to identify the parameters whose uncertainty has the largestimpact on the variability of a quantity of interest (output of the model). One of thestatistical tools used to quantify the influence of each input variable on the output isthe Sobol sensitivity index. We consider the statistical estimation of this index from afinite sample of model outputs: we present two estimators and state a central limittheorem for each. We show that one of these estimators has an optimal asymptotic variance.We also generalize our results to the case where the true output is not observable, and isreplaced by a noisy version.

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincaré and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears aslimit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process isnon-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral withrespect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.

We consider a diffusion process X which is observed at times i/nfor i = 0,1,...,n, each observation being subject to a measurementerror. All errors are independent and centered Gaussian with knownvariance pn . There is an unknown parameter within the diffusioncoefficient, to be estimated. In this first paper thecase when X is indeed a Gaussian martingale is examined: we can provethat the LAN property holds under quite weak smoothness assumptions,with an explicit limiting Fisher information. What is perhaps the mostinteresting is the rate at which this convergence takes place:it is $1/\sqrt{n}$ (as when there is no measurement error) when pn goes fastenough to 0, namely npn is bounded. Otherwise, and provided thesequence pn itself is bounded, the rate is (pn / n) 1/4. Inparticular if pn = p does not depend on n, we get a rate n -1/4.

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form|x|α , α ∈ [1/2,1). In that case, we study the rate of convergence of asymmetrized version of the Euler scheme. This symmetrized version iseasy to simulate on a computer. We prove its strong convergence and obtain the same rate ofconvergence as when the coefficients are Lipschitz.

Stein's method is used to prove approximations in total variation to thedistributions of integer valued random variables by (possibly signed)compound Poisson measures. For sums of independent random variables,the results obtained are very explicit, and improve upon earlierwork of Kruopis (1983) and Čekanavičius (1997);coupling methods are used to derive concrete expressions for the errorbounds. An example is given to illustrate the potential for applicationto sums of dependent random variables.

We consider a diffusion process X which is observed at times i/nfor i = 0,1,...,n, each observation being subject to a measurementerror. All errors are independent and centered Gaussian with knownvariance pn . There is an unknown parameter to estimate within thediffusion coefficient. In this second paper weconstruct estimators which are asymptotically optimal when theprocess X is a Gaussian martingale, and we conjecture that they arealso optimal in the general case.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2, we focus on the log normal case in dimension 2.

Let X be a random element in a metric space (F,d), and let Y be a random variable with value 0 or 1. Y iscalled the class, or the label, of X. Let (Xi,Yi)1 ≤ i ≤ n be an observed i.i.d. sample having the same law as (X,Y). The problem of classification is topredict the label of a new random element X. The k-nearestneighbor classifier is the simple following rule: look atthe k nearest neighbors of X in the trial sample and choose 0 or 1 for its labelaccording to the majority vote. When $({\cal F},d)=(\mathbb{R}^d,||.||)$ ,Stone (1977) proved in 1977 the universal consistency of this classifier:its probability of error converges to the Bayes error, whatever thedistribution of (X,Y). We show in this paper that this result is nolonger valid in general metric spaces. However, if (F,d) isseparable and if some regularity condition is assumed, then thek-nearest neighbor classifier is weakly consistent.

The paper is motivated by the stochastic comparison of the reliabilityof non-repairable k-out-of-n systems.The lifetime of such a system with nonidentical components is compared with the lifetime of a system withidentical components.Formally the problem is as follows. Let Ui,i = 1,...,n, be positiveindependent random variables with common distribution F.For λi > 0 and µ > 0, let considerXi = Ui/λi and Yi = Ui/µ, i = 1,...,n. Remark that this is no more than a change of scale for each term.For k ∈ {1,2,...,n}, let us define Xk:n to be the kth order statistics of the random variables X1,...,Xn , andsimilarly Yk:n to be the kth order statistics ofY1,...,Yn .If Xi,i = 1,...,n, are the lifetimes of the components of an+1-k-out-of-n non-repairable system, then Xk:n is thelifetime of the system.In this paper, we give for a fixed k a sufficient condition forXk:n ≥st Yk:n where st is the usual ordering for distributions.In the Markovian case (all components have an exponential lifetime), wegive a necessary and sufficient condition.We prove that Xk:n is greater that Yk:n according to the usualstochastic ordering if and only if \[\left( \begin{array}{c} n k \end{array}\right){\mu}^k\geq \sum_{1\leqi_1<i_2<...<i_k\leq n}\lambda_{i_1}\lambda_{i_2}...\lambda_{i_k}.\]

Given an n-sample from some unknown density f on [0,1], it is easy to construct anhistogram of the data based on some given partition of [0,1], but not so much is knownabout an optimal choice of the partition, especially when the data set is not large, even ifone restricts to partitions into intervals of equal length. Existing methods are either rulesof thumbs or based on asymptotic considerations and often involve some smoothnessproperties of f. Our purpose in this paper is to give an automatic, easy to program andefficient method to choose the number of bins of the partition from the data. It is based on boundson the risk of penalized maximum likelihood estimators due to Castellan and heavy simulationswhich allowed us to optimize the form of the penalty function. These simulations show that themethod works quite well for sample sizes as small as 25.

This paper is concerned with the problem of simulation of (Xt)0≤t≤T , thesolution of a stochastic differential equation constrained by some boundary conditions in a smooth domainD: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneouslyreflecting in an oblique direction. Given N discretization times equally spaced on the interval [0,T],we propose new discretization schemes: they are fully implementable and provide a weak error of orderN -1 under some conditions. The construction of these schemes is based on a natural principle of localapproximation of the domain into a half space, for which efficient simulations are available.

We consider the problem of estimating an unknown regression function when the design is random with values in $\mathbb{R}^k$. Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability. For the so-defined estimator, we establish nonasymptotic risk bounds that can be related to oracle inequalities. As a consequence of these, we show that our estimator possesses adaptive properties in the minimax sense over large families of Besov balls Bα,l,∞(R) with R>0, l ≥ 1 and α > α1 where α1 is a positive number satisfying 1/l - 1/2 ≤ α1 < 1/l. We also study the particular case where the regression function is additive and then obtain an additive estimator which converges at the same rate as it does when k=1.


In this paper, we address the problem of testing hypotheses using maximum likelihood statistics in non identifiable models. We derive the asymptotic distribution under very general assumptions.The key idea is a local reparameterization, depending on the underlying distribution, which is called locally conic. This method enlights how the general model induces the structure of the limiting distribution in terms of dimensionality of some derivative space. We present various applications of the theory. The main application is to mixture models. Under very general assumptions, we solve completely the problem of testing the size of the mixture using maximum likelihood statistics. We derive the asymptotic distribution of the maximum likelihood statistic ratio which takes an unexpected form.

Variance reduction has always been a central issue in Monte Carlo experiments. Population Monte Carlo can be used to this effect, in that a mixture of importance functions, called a D-kernel, can be iterativelyoptimized to achieve the minimum asymptotic variance for a function of interest among all possible mixtures.The implementation of this iterative scheme is illustrated for the computation of the price of a Europeanoption in the Cox-Ingersoll-Ross model. A Central Limit theorem as well as moderate deviations are established for the D-kernel Population Monte Carlo methodology.


We prove a full large deviations principle, in the scale N2, for the empirical measure of the eigenvalues of an N x N (non self-adjoint) matrix composed of i.i.d. zero mean random variables with variance N-1. The (good) rate function which governs this rate function possesses as unique minimizer the circular law, providing an alternative proof of convergence to the latter. The techniques are related to recent work by Ben Arous and Guionnet, who treat the self-adjoint case. A crucial role is played by precise determinant computations due to Edelman and to Lehmann and Sommers.

A multidimensional version of the results of Komlos, Major and Tusnady for the Gaussian approximation of the sequence of successive sums of independent random vectors with finiteexponential moments is obtained.

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process betweenthe successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/orcentering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f. As for the associated central limit theorem,one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.