Elodie Brunel

Adaptive Estimation of Hazard Rate with Censored Data

In this article, we study hazard projection estimators under right-censoring of the observations. We consider projection spaces generated by trigonometric or piecewise polynomials. First, we derive bounds on the MISE of the estimators. These bounds imply that the estimator reaches the standard optimal rate associated with the regularity of the hazard function, provided that the dimension of the projection space is relevantly chosen. Then we provide an adaptive procedure leading to an automatic choice of this dimension via a penalized minimum contrast estimator. Our procedure is based on a data-driven random penalty function. The resulting estimator automatically reaches the optimal rate.

CUMULATIVE DISTRIBUTION FUNCTION ESTIMATION UNDER INTERVAL CENSORING CASE 1.

We consider projection methods for the estimation of cumulative distribu- tion function under interval censoring, case 1. Such censored data also known as cur- rent status data, arise when the only information available on the variable of interest is whether it is greater or less than an observed random time. Two types of adaptive estima- tors are investigated. The rst one is a two-step estimator built as a quotient estimator. The second estimator results from a mean square regression contrast. Both estimators are proved to achieve automatically the standard optimal rate associated with the un- known regularity of the function, but with some restriction for the quotient estimator. Simulation experiments are presented to illustrate and compare the methods.

Non-asymptotic adaptive prediction in functional linear models

Functional linear regression has recently attracted considerable interest. Many works focus on asymptotic inference. In this paper we consider in a non asymptotic framework a simple estimation procedure based on functional Principal Regression. It revolves in the minimization of a least square contrast coupled with a classical projection on the space spanned by the m first empirical eigenvectors of the covariance operator of the functional sample. The novelty of our approach is to select automatically the crucial dimension m by minimization of a penalized least square contrast. Our method is based on model selection tools. Yet, since this kind of methods consists usually in projecting onto known non-random spaces, we need to adapt it to empirical eigenbasis made of data-dependent-hence random-vectors. The resulting estimator is fully adaptive and is shown to verify an oracle inequality for the risk associated to the prediction error and to attain optimal minimax rates of convergence over a certain class of ellipsoids. Our strategy of model selection is finally compared numerically with cross-validation.

Penalized contrast estimation in functional linear models with circular data

Our aim is to estimate the unknown slope function in the functional linear model when the response Y is real and the random function X is a second-order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. Our approach provides a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real example that illustrates how the procedure works in practice.

Cross-validated density estimates based on Kullback–Leibler information

The convergence of measure estimates in the sense of Kullback–Leibler divergence is required in many applications in decision and information theory. Recently, modified histograms have been shown to have good properties with respect to information divergences. For these estimates deterministic optimal bandwidths have been given, but no automatic smoothing procedure has been shown to be asymptotically optimal. In the present article, we consider the Kullback–Leibler cross-validation method for selecting the bin width of modified histograms. We analyze the behavior of the Kullback–Leibler divergence and of its expectation and prove that the cross-validated estimate is asymptotically optimal with respect to the Kullback–Leibler divergence.

Conditional mean residual life estimation

In this paper, we consider the problem of nonparametric mean residual life (MRL) function estimation in presence of covariates. We propose a contrast that provides estimators of the bivariate conditional MRL function, when minimised over different collections of linear finite-dimensional function spaces. Then we describe a model selection device to select the best estimator among the collection, in the mean integrated squared error sense. A non-asymptotic oracle inequality is proved for the estimator, which both ensures the good finite sample performances of the estimator and allows us to compute asymptotic rates of convergence. Lastly, examples and simulation experiments illustrate the method, together with a short real data study.