HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short-term dynamic models, recent works considered more complex long-term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation-maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long-term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2-ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between-patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. d-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max U -statistic, hence the maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.

Ill-posed problem, Singular value decomposition, Iterative regularization methods, Model selection, Descent methods, 62F03, 62E17, 62P25,

In this article we consider the problem of active learning in the regression setting. That is, choosing an optimal sampling scheme for the regression problem simultaneously with that of model selection. We consider a batch type approach and an on-line approach adapting algorithms developed for the classification problem. Our main tools are concentration-type inequalities which allow us to bound the supreme of the deviations of the sampling scheme corrected by an appropriate weight function.

No abstract