Oracle inequalities for the Lasso in the high-dimensional Aalen multiplicative intensity model

In a general counting process setting, we consider the problem of obtaining a prognostic on the survival time adjusted on covariates in high-dimension. Towards this end, we construct an estimator of the whole conditional intensity. We estimate it by the best Cox proportional hazards model given two dictionaries of functions. The first dictionary is used to construct an approximation of the logarithm of the baseline hazard function and the second to approximate the relative risk. We introduce a new data-driven weighted Lasso procedure to estimate the unknown parameters of the best Cox model approximating the intensity. We provide non-asymptotic oracle inequalities for our procedure in terms of an appropriate empirical Kullback divergence. Our results rely on an empirical Bernstein's inequality for martingales with jumps and properties of modified self-concordant functions

Adaptive kernel estimation of the baseline function in the Cox model, with high-dimensional covariates

The aim of this article is to propose a novel kernel estimator of the baseline function in a general high-dimensional Cox model, for which we derive non-asymptotic rates of convergence. To construct our estimator, we first estimate the regression parameter in the Cox model via a Lasso procedure. We then plug this estimator into the classical kernel estimator of the baseline function, obtained by smoothing the so-called Breslow estimator of the cumulative baseline function. We propose and study an adaptive procedure for selecting the bandwidth, in the spirit of Gold-enshluger and Lepski (2011). We state non-asymptotic oracle inequalities for the final estimator, which reveal the reduction of the rates of convergence when the dimension of the covariates grows

Estimation and variable selection in a joint model of survival times and longitudinal outcomes with random effects

This paper considers a joint survival and mixed-effects model to explain the survival time from longitudinal data and high-dimensional covariates. The longitudinal data is modeled using a nonlinear effects model, where the regression function serves as a link function incorporated into a Cox model as a covariate. In that way, the longitudinal data is related to the survival time at a given time. Additionally, the Cox model takes into account the inclusion of high-dimensional covariates. The main objectives of this research are two-fold: first, to identify the relevant covariates that contribute to explaining survival time, and second, to estimate all unknown parameters of the joint model. For that purpose, we consider the maximization of a Lasso penalized likelihood. To tackle the optimization problem, we implement a pre-conditioned stochastic gradient to handle the latent variables of the nonlinear mixed-effects model associated with a proximal operator to manage the non-differentiability of the penalty. We provide relevant simulations that showcase the performance of the proposed variable selection and parameters' estimation method in the joint modeling of a Cox and logistic model

On the nonparametric inference of coefficients of self-exciting jump-diffusion

In this paper, we consider a one-dimensional diffusion process with jumps driven by a Hawkes process. We are interested in the estimations of the volatility function and of the jump function from discrete high-frequency observations in a long time horizon which remained an open question until now. First, we propose to estimate the volatility coefficient. For that, we introduce a truncation function in our estimation procedure that allows us to take into account the jumps of the process and estimate the volatility function on a linear subspace of L2(A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator, ensuring its consistency, and then we study an adaptive estimator w.r.t. the regularity. Then, we define an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. We also establish a bound for the empirical risk for the non-adaptive estimators of this sum, the convergence rate up to the regularity of the true function, and an oracle inequality for the final adaptive estimator.Finally, we give a methodology to recover the jump function in some applications. We conduct a simulation study to measure our estimators' accuracy in practice and discuss the possibility of recovering the jump function from our estimation procedure

Efficient preconditioned stochastic gradient descent for estimation in latent variable models

Latent variable models are powerful tools for modeling complex phenomena involving in particular partially observed data, unobserved variables or underlying complex unknown structures. Inference is often difficult due to the latent structure of the model. To deal with parameter estimation in the presence of latent variables, well-known efficient methods exist, such as gradient-based and EM-type algorithms, but with practical and theoretical limitations. In this paper, we propose as an alternative for parameter estimation an efficient preconditioned stochastic gradient algorithm. Our method includes a preconditioning step based on a positive definite Fisher information matrix estimate. We prove convergence results for the proposed algorithm under mild assumptions for very general latent variables models. We illustrate through relevant simulations the performance of the proposed methodology in a nonlinear mixed effects model and in a stochastic block model

Estimation pour les processus de comptage avec beaucoup de covariables

We consider the problem of estimating the intensity of a counting process adjusted on high-dimensional covariates. We propose two different approaches. First, we consider a non-parametric intensity function and estimate it by the best Cox proportional hazards model given two dictionaries of functions. The first dictionary is used to construct an approximation of the logarithm of the baseline hazard function and the second to approximate the relative risk. In this high-dimensional setting, we consider the Lasso procedure to estimate simultaneously the unknown parameters of the best Cox model approximating the intensity. We provide non-asymptotic oracle inequalities for the resulting Lasso estimator. In a second part, we consider an intensity that rely on the Cox model. We propose two two-step procedures to estimate the unknown parameters of the Cox model. Both procedures rely on a first step which consists in estimating the regression parameter in high-dimension via a Lasso procedure. The baseline function is then estimated either via model selection or by a kernel estimator with a bandwidth selected by the Goldenshluger and Lepski method. We establish non-asymptotic oracle inequalities for the two resulting estimators of the baseline function. We conduct a comparative study of these estimators on simulated data, and finally, we apply the implemented procedure to a real dataset on breast cancer.Nous cherchons à estimer l'intensité de sauts d'un processus de comptage en présence d'un grand nombre de covariables. Nous proposons deux approches. D'abord, nous considérons une intensité non-paramétrique et nous l'estimons par le meilleur modèle de Cox étant donné deux dictionnaires de fonctions. Le premier dictionnaire est utilisé pour construire une approximation du logarithme du risque de base et le second pour approcher le risque relatif. Nous considérons une procédure Lasso, spécifique à la grande dimension, pour estimer simultanément les deux paramètres inconnus du meilleur modèle de Cox approchant l'intensité. Nous prouvons des inégalités oracles non-asymptotiques pour l'estimateur Lasso obtenu.Dans une seconde partie, nous supposons que l'intensité satisfait un modèle de Cox. Nous proposons deux procédures en deux étapes pour estimer les paramètres inconnus du modèle de Cox. La première étape est commune aux deux procédures, il s'agit d'estimer le paramètre de régression en grande dimension via une procédure Lasso. Le risque de base est ensuite estimé soit par sélection de modèles, soit par un estimateur à noyau avec une fenêtre choisie par la méthode de Goldenshluger et Lepski. Nous établissons des inégalités oracles non-asymptotiques pour les deux estimateurs du risque de base ainsi obtenus. Nous menons une étude comparative de ces estimateurs sur des données simulées, et enfin, nous appliquons les procédures implémentées à une base de données sur le cancer du sein

Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process

International audienceWe consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. This article is dedicated to the study of a nonparametric estimator of the drift coefficient of this original process. We construct estimators based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. The proposed nonparametric estimator is built from a least squares contrast procedure on subspace spanned by trigonometric basis vectors. We obtain adaptive results that are comparable with the one obtained in the nonparametric regression context. We finally conduct a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator

Exponential ergodicity for diffusions with jumps driven by a Hawkes process

International audienceIn this paper, we introduce a new class of processes which are diffusions with jumps driven by a multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process we establish conditions for the positive Harris recurrence of the couple (X, Y), where X denotes the diffusion process and Y the piecewise deterministic Markov process (PDMP) defining the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provide sufficient conditions under which the process is exponentially β−mixing

Estimation and variable selection in joint models of survival times and longitudinal data

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Estimation and variable selection in a joint model of survival times and longitudinal outcomes with random effects.

This paper considers a joint survival and mixed-effects model to explain the survival time from longitudinal data and high-dimensional covariates. The longitudinal data is modeled using a nonlinear effects model, where the regression function serves as a link function incorporated into a Cox model as a covariate. In that way, the longitudinal data is related to the survival time at a given time. Additionally, the Cox model takes into account the inclusion of high-dimensional covariates. The main objectives of this research are two-fold: first, to identify the relevant covariates that contribute to explaining survival time, and second, to estimate all unknown parameters of the joint model. For that purpose, we consider the maximization of a Lasso penalized likelihood. To tackle the optimization problem, we implement a pre-conditioned stochastic gradient to handle the latent variables of the nonlinear mixed-effects model associated with a proximal operator to manage the non-differentiability of the penalty. We provide relevant simulations that showcase the performance of the proposed variable selection and parameters' estimation method in the joint modeling of a Cox and logistic model