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Guillaume MAILLARD (Luxembourg) – “Robust density estimation in total variation under a shape constraint”

March 20 @ 2:00 pm - 3:15 pm

Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 20th of March 2023
Place: Room 3001 + ZOOM


Guillaume MAILLARD (Luxembourg) – “Robust density estimation in total variation under a shape constraint


Code secret : 387963



We solve the problem of estimating the distribution of presumed i.i.d observations for the total variation loss, under the assumption that the underlying density satisfies a shape constraint of the following type: monotonicity on an interval, unimodality, convexity/concavity on an interval or log-concavity. Our estimator is well-defined even in cases where the MLE isn’t, such as for unimodal densities with unknown mode. Moreover, it possesses similar optimality properties, with regard to some global rates of convergence, as the MLE does when it exists. Like the MLE, it also enjoys some adaptation properties with respect to some specific target densities in the model, for which our estimator is proven to converge at a parametric rate. Unlike the MLE, in general, our estimator is robust, not only with respect to model mis-specification, but also to contamination, the presence of outliers among the dataset and the equidistribution assumption.

Our main result on the risk of the estimator takes the form of an exponential deviation inequality which is non-asymptotic and involves explicit numerical constants. We deduce from it several global rates of convergence, including some bounds for the minimax L1 risks over sets of convex or monotone densities. These bounds derive from specific results on the approximation of densities which are monotone, convex, concave or log-concave. Such results may be of independent interest.

The presentation is based on joint work with Yannick Baraud and Hélène Halconruy.