This paper introduces a computationally tractable density estimatorthat has the same asymptotic variance as the classical Nadaraya-Watsondensity estimator but whose asymptotic bias is zero. We achieve this resultusing a two stage estimator that applies a multiplicative bias correctionto an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator.

In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which one can visualize repetitions of subwords(seen as suffixes of subsequences). The CGR-tree turns a sequence of letters into a Digital Search Tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height, the insertion depth for DST built from independent successive random sequences having the same distribution. Here the successive inserted words are strongly dependent. We give the asymptotic behaviour of the insertion depth and the length of branches for the CGR-tree obtained from the suffixes of a reversed i.i.d. or Markovian sequence.This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained.

This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k,m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m. We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models).

Consider a random environment in ${\mathbb Z}^d$ given by i.i.d. conductances.In this work, we obtain tail estimates for the fluctuations about themean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environmentsand the spectral gap of the random walk in a finite cube.

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping $\mathcal{X}$ into $\mathbb{R}$ .We consider a non Gaussian regression framework with deterministic design points, and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Given a collection of partitions of $\mathcal{X}$ , with possibly exponential complexity,and the corresponding collection of piecewise constant estimators, we propose a penalized least squares criterion which selects a partition whose associated estimator performs approximately as well as the best one, in the sense that its quadratic risk is close to the infimum of the risks. The risk bound we provide is non asymptotic.

We consider a failure hazard function,conditional on a time-independent covariate Z,given by $\eta_{\gamma^0}(t)f_{\beta^0}(Z)$ . The baseline hazardfunction $\eta_{\gamma^0}$ and the relative risk $f_{\beta^0}$ both belong to parametricfamilies with $\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb{R}^{m+p}$ . The covariate Z has an unknown density and is measured with an error through anadditive error model U = Z + ε where ε is a random variable, independent from Z, withknown density $f_\varepsilon$ .We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where X i isthe minimum between the failure time and the censoring time,and D i is the censoring indicator.Using least square criterion and deconvolution methods, we propose a consistent estimator of θ 0 using the observationsn-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its riskwhich depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as afunction of z, and we derive sufficient conditionsfor the $\sqrt{n}$ -consistency.We give detailed examples consideringvarious type of relative risks $f_{\beta}$ and various types of errordensity $f_\varepsilon$ . In particular, in the Cox model and inthe excess risk model, the estimator of θ 0 is $\sqrt{n}$ -consistent and asymptotically Gaussianregardless of the form of $f_\varepsilon$ .

We present two approaches for linear prediction of long-memory time series. The first approach consists in truncating the Wiener-Kolmogorov predictor by restricting the observations to the last k terms, which are the only available data in practice. We derive the asymptotic behaviour of the mean-squared error as k tends to +∞. The second predictor is the finite linear least-squares predictor i.e.  the projection of the forecast value on the last k observations. It is shown that these two predictors converge to the Wiener Kolmogorov predictor at the same rate k -1.

In this paper we obtain root-n consistency and functional central limittheorems in weighted L 1-spaces for plug-in estimators of thetwo-step transition density in the classical stationary linear autoregressivemodel of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-basedkernel estimator for the unknown innovation densityimproves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit thefact that the innovations have mean zero.If an efficient estimator for the autoregression parameter is used,then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes.

As in preceding papers inwhich we studied the limits of penalized 1-dimensional Wienermeasures with certain functionals Γt, we obtain here theexistence of the limit, as t → ∞, of d-dimensional Wienermeasures penalized by a function of the maximum up to time t ofthe Brownian winding process (for d = 2), or in {d}≥ 2dimensions for Brownian motionprevented to exit a cone before time t. Various extensions of these multidimensional penalisations arestudied, and the limit laws are described. Throughout this paper, the skew-product decomposition ofd-dimensional Brownian motion plays an important role.

Even for a well-trained statistician the construction of a histogramfor a given real-valued data set is a difficult problem. It is evenmore difficult to construct a fully automatic procedure whichspecifies the number and widths of the bins in a satisfactory mannerfor a wide range of data sets. In this paper we compare severalhistogram construction procedures by means of a simulationstudy. The study includes plug-in methods, cross-validation,penalized maximum likelihood and the taut string procedure. Their performance on different test beds is measured by their ability to identify the peaks of an underlying density as well as by Hellinger distance.

We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t 1 < t 2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i ≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr -optimal quantizers of an $\mathbb{R}^d$ -valued random vector $X \in L^r(\mathbb{P})$ defined in the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with distribution $\mathbb{P}_{X} = P$ . To be precise, we investigate the Ls -quantization rate of sequences $\alpha_n^{\theta,\mu} = \mu + \theta(\alpha_n-\mu)=\{\mu + \theta(a-\mu), \ a \in \alpha_n \}$ when $\theta \in \mathbb{R}_{+}^{\star}, \mu \in \mathbb{R}, s \in (0,r)$ or s ∈ (r, +∞) and $X \in L^s(\mathbb{P})$ . We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls -rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (αθ*,µ*)n≥1 also satisfies the so-called Ls -empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically Ls -optimal. In both cases the sequence (αθ*,µ*)n≥1 is incredibly close to Ls -optimality. However we show (see Rem. 5.4) that this last sequence is not Ls -optimal (e.g. when s = 2, r = 1) for the exponential distribution.

We propose some construction of enhanced Gaussian processes usingKarhunen-Loeve expansion. We obtain a characterization and somecriterion of existence and uniqueness. Using rough-path theory, wederive some Wong-Zakai Theorem.

Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$ , n ≥ 1, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are i.i.d., centered random elements in some separable Hilbert space $\mathbb{H}$ and the a i 's are bounded linear operators $\mathbb{H}\to \mathbb{H}$ , with $\sum_{i\ge 0}\lVert a_i\rVert<\infty$ . We investigate functional central limit theorem for $\xi_n$ in the Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$ of functions $x:[0,1]\to\mathbb{H}$ such that ||x(t + h) - x(t)|| = o(p(h)) uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1 with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We obtain the $\mathrm{H}^o_\rho(\mathbb{H})$ weak convergence of $\xi_n$ to some $\mathbb{H}$ valued Brownian motion under the optimal assumption that for any c>0, $tP(\lVert \epsilon_0\rVert>ct^{1/2}\rho(1/t))=o(1)$ when t tends to infinity, subject to some mild restriction on L in the boundary case α = 1/2. Our result holds in particular with the weight functions p(h) = h1/2 lnβ(1/h), β > 1/2>.

The aim is to study the asymptotic behavior of estimators and testsfor the components of identifiable finite mixture models ofnonparametric densities with a known number of components.Conditions for identifiability of the mixture components andconvergence of identifiable parameters are given.The consistency and weak convergence of the identifiable parametersand test statistics are presented for several models.

Consider testing whether F = F0 for a continuous cdf on R = (-∞,∞)and for a random sample X 1,..., X n from F.We derive expansions of the associated asymptotic power basedon the Cramer-von Mises, Kolmogorov-Smirnov and Kuiper statistics. We provide numerical illustrations using a double-exponential example with a shifted alternative.

This paper deals with the likelihood ratio test (LRT) for testing hypotheseson the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRTstatisticsunder some conditions that are proved to be almost necessary.A detailed solution is given for two testing problems: thetest of a single distribution against any mixture, with application to Gaussian, Poisson andbinomial distributions; the test of the number of populations in afinite mixture with or without structural parameter.

Let a finite alphabet Ω. We consider a sequence of letters from Ωgenerated by a discrete time semi-Markov process $\{Z_{\gamma};\ \gamma\in \xN \}.$ Wederive the probability of a word occurrence in the sequence. We also obtain resultsfor the mean and variance of the number of overlapping occurrences of a word in afinite discrete time semi-Markov sequence of letters under certain conditions.

We study two systems that are based on sums of weakly dependentBernoulli random variables that take values ± 1 with equalprobabilities. We show that already one step of the so-calledsoft decision parallel interference cancellation, used in the thirdgeneration of mobile telecommunication CDMA, is able to considerablyincrease the number of users such a system can host. We alsoconsider a variant of the well-known Hopfield model of neuralnetworks. We show that this variant proposed by Amari and Yanai[CITE] has a larger storage capacity than the original model. Both situations lead to the question of the moderatedeviations behavior of a sum of weakly dependent Bernoulli randomvariables. We prove a moderate deviations principle for such a sumon the appropriate scale.

This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators,where we may use pattern functions or wavelets, and penalized maximumlikelihood estimators. In all these cases, we get oracle inequalities. In theformer we also have a polynomial rate of convergence for the non-parametricproblem. We finish the paper with applications of similar ideas to thecalibration of a photocounter, a measurement apparatus counting the number ofphotons in a beam. Here the mathematical problem reduces similarly to anon-parametric missingdata problem. We again get oracle inequalities, and better speed if thephotocounter is good.