Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes

A piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. We focus on the nonparametric estimation problem of the jump rate for such a stochastic model observed within a long time interval under an ergodicity condition. We introduce an uncountable class (indexed by the deterministic flow) of recursive kernel estimates of the jump rate and we establish their strong pointwise consistency as well as their asymptotic normality. We propose to choose among this class the estimator with the minimal variance, which is unfortunately unknown and thus remains to be estimated. We also discuss the choice of the bandwidth parameters by cross-validation methods.Comment: 36 pages, 18 figure

Modélisation de réseaux de régulation de gènes par processus déterministes par morceaux

6 pagesInternational audienceThe molecular evolution in a gene regulatory network is classically modeled by Markov jump processes. However, the direct simulation of such models is extremely time consuming. Indeed, even the simplest Markovian model, such as the production module of a single protein involves tens of variables and biochemical reactions and an equivalent number of parameters. We study the asymptotic behavior of multiscale sto- chastic gene networks using weak limits of Markov jump processes. The results allow us to propose new models with reduced execution times. In a new article, we have shown that, depending on the time and concentration scales of the system, the Markov jump processes could be approximated by piecewise deterministic processes. We give some applications of our results for simple gene networks (Cook's model and Lambda-phage model)

Package 'armada' : A Statistical Methodology to Select Covariates in High-Dimensional Data under Dependence

An R package, available on the CRAN. A Statistical Methodology to Select Covariates in High-Dimensional Data under Dependenc

Estimation non paramétrique optimale du taux de saut d'un processus markovien déterministe par morceaux

International audienceUn processus markovien déterministe par morceaux est un processus stochastique dont la trajectoire est décrite par une équation différentielle perturbée par des sauts aléatoires en des instants aléatoires. Nous nous intéressons à l'estimation du taux de saut d'un tel processus observé en temps long sous une hypothèse d'ergodicité. Nous introduisons une classe d'estimateurs non paramétriques consistants et asymptotiquement gaussiens. Nous proposons de choisir l'estimateur de variance minimale, variance qui est elle-même à estimer

The revisited knockoffs method for variable selection in L1L_1-penalised regressions.

We consider the problem of variable selection in regression models. In particular, we are interested in selecting explanatory covariates linked with the response variable and we want to determine which covariates are relevant, that is which covariates are involved in the model. In this framework, we deal with $L_1$-penalised regression models. To handle the choice of the penalty parameter to perform variable selection, we develop a new method based on the knockoffs idea. This revisited knockoffs method is general, suitable for a wide range of regressions with various types of response variables. Besides, it also works when the number of observations is smaller than the number of covariates and gives an order of importance of the covariates. Finally, we provide many experimental results to corroborate our method and compare it with other variable selection methods

Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes

International audienceA piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. We focus on the nonparametric estimation problem of the jump rate for such a stochastic model observed within a long time interval under an ergodicity condition. We introduce an uncountable class (indexed by the deterministic flow) of recursive kernel estimates of the jump rate and we establish their strong pointwise consistency as well as their asymptotic normality. We propose to choose among this class the estimator with the minimal variance, which is unfortunately unknown and thus remains to be estimated. We also discuss the choice of the bandwidth parameters by cross-validation methods

Graph estimation for Gaussian data zero-inflated by double truncation

We consider the problem of graph estimation in a zero-inflated Gaussian model. In this model, zero-inflation is obtained by double truncation (right and left) of a Gaussian vector. The goal is to recover the latent graph structure of the Gaussian vector with observations of the zero-inflated truncated vector. We propose a two step estimation procedure. The first step consists in estimating each term of the covariance matrix by maximising the corresponding bivariate marginal log-likelihood of the truncated vector. The second one uses the graphical lasso procedure to estimate the precision matrix sparsity, which encodes the graph structure. We then state some theoretical convergence results about the convergence rate of the covariance matrix and precision matrix esti-mators. These results allow us to establish consistency of our procedure with respect to graph structure recovery. We also present some simulation studies to corroborate the efficiency of our procedure

Penalized ordinal logistic regression using cumulative logits

International audienc