Robust density estimation with the L1-loss. Applications to the estimation of a density on the line satisfying a shape constraint

E-print/Working paper (2022)

We solve the problem of estimating the distribution of presumed i.i.d.\ observations for the total variation loss. Our approach is based on density models and is versatile enough to cope with many ... [more ▼]

We solve the problem of estimating the distribution of presumed i.i.d.\ observations for the total variation loss. Our approach is based on density models and is versatile enough to cope with many different ones, including some density models for which the Maximum Likelihood Estimator (MLE for short) does not exist. We mainly illustrate the properties of our estimator on models of densities on the line that satisfy a shape constraint. We show that it possesses some similar optimality properties, with regard to some global rates of convergence, as the MLE does when it exists. 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 parametric rate. More important is the fact that our estimator is robust, not only with respect to model misspecification, but also to contamination, the presence of outliers among the dataset and the equidistribution assumption. This means that the estimator performs almost as well as if the data were i.i.d.\ with density $p$ in a situation where these data are only independent and most of their marginals are close enough in total variation to a distribution with density $p$. {We also show that our estimator converges to the average density of the data, when this density belongs to the model, even when none of the marginal densities belongs to it}. 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 the sets of concave and log-concave densities. These bounds derive from some specific results on the approximation of densities which are monotone, convex, concave and log-concave. Such results may be of independent interest. [less ▲]

Tests and estimation strategies associated to some loss functions

in Probability Theory and Related Fields (2021), 180(3), 799-846

We consider the problem of estimating the joint distribution of n independent random variables. Given a loss function and a family of candidate probabilities, that we shall call a model, we aim at ... [more ▼]

We consider the problem of estimating the joint distribution of n independent random variables. Given a loss function and a family of candidate probabilities, that we shall call a model, we aim at designing an estimator with values in our model that possesses good estimation properties not only when the distribution of the data belongs to the model but also when it lies close enough to it. The losses we have in mind are the total variation, Hellinger, Wasserstein and L_p-distances to name a few. We show that the risk of our estimator can be bounded by the sum of an approximation term that accounts for the loss between the true distribution and the model and a complexity term that corresponds to the bound we would get if this distribution did belong to the model. Our results hold under mild assumptions on the true distribution of the data and are based on exponential deviation inequalities that are non-asymptotic and involve explicit constants. Interestingly, when the model reduces to two distinct probabilities, our procedure results in a robust test whose errors of first and second kinds only depend on the losses between the true distribution and the two tested probabilities. [less ▲]

Estimating the number of infected persons

E-print/Working paper (2020)

The aim of this paper is to provide a confidence interval on the number of infected persons by COVID-19 within the population from the number of deaths reported in the hospitals and the mortality rate ... [more ▼]

The aim of this paper is to provide a confidence interval on the number of infected persons by COVID-19 within the population from the number of deaths reported in the hospitals and the mortality rate (that is assumed to be known). [less ▲]

Can we trust L2-criteria and L2-losses?

in Journal de la Société Française de Statistique (2019), 160(3),

A new method for estimation and model selection: $\rho$-estimation

in Inventiones Mathematicae (2017), 207(2), 425--517

Bounding the expectation of the supremum of an empirical process over a (weak) VC-major class

in Electronic Journal of Statistics (2016), 10(2), 1709--1728

Estimating composite functions by model selection

in Ann. Inst. Henri Poincaré Probab. Stat. (2014), 50(1), 285--314

Estimation of the density of a determinantal process

in Confluentes Mathematici (2013), 5(1), 3--21

A Bernstein-type inequality for suprema of random processes with applications to model selection in non-Gaussian regression

in Bernoulli (2010), 16(4), 1064--1085

Estimating the intensity of a random measure by histogram type estimators

in Probab. Theory Related Fields (2009), 143(1-2), 239--284