Christophe Abraham

Simple estimation of the mode of a multivariate density

The authors consider an estimate of the mode of a multivariate probability density using a kernel estimate drawn from a random sample. The estimate is defined by maximizing the kernel estimate over the set of sample values. The authors show that this estimate is strongly consistent and give an almost sure rate of convergence. This rate depends on the sharpness of the density near the true mode, which is measured by a peak index.

Asymptotic global robustness in Bayesian decision theory

In Bayesian decision theory, it is known that robustness with respect to the loss and the prior can be improved by adding new observations. In this article we study the rate of robustness improvement with respect to the number of observations n. Three usual measures of posterior global robustness are considered: the (range of the) Bayes actions set derived from a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class. We show that the rate of convergence of the first measure of robustness is √n, while it is n for the other measures under reasonable assumptions on the class of loss functions. We begin with the study of two particular cases to illustrate our results.

Asymptotic properties of posterior distributions derived from misspecified models

Abstract We investigate the asymptotic properties of posterior distributions when the model is misspecified, i.e. it is contemplated that the observations x 1 ,…, x n might be drawn from a density in a family {h σ , σ∈Θ} where Θ⊂ R d , while the actual distribution of the observations may not correspond to any of the densities h σ . A concentration property around a fixed value of the parameter is obtained as well as concentration properties around the maximum likelihood estimate. To cite this article: C. Abraham, B. Cadre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 495–498.

Analytic approximation of the interval of Bayes actions derived from a class of loss functions

Due to a lack of time or limited information, a precise determination of the loss function, and the computation of the associated Bayes action is not always possible. For these reasons, statisticians use mathematically tractable losses instead of subjective ones. We study the legitimacy of such lossesl0 with specific properties, by first constructing a class aroundl0 which contains all plausible subjective losses, and then computing the setAπ of Bayes actions associated with this class. We derive an analytic approximation toAπ. We apply these results whenlo is the quadratic, LIXEX and entropy loss. Finally, we investigate robustness characteristics using the analytic expression ofAπ.