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We solve the problem of estimating the distribution of presumed i.i.d.observations for the total variation loss. Our approach is based on densitymodels and is versatile enough to cope with many different ones, including somedensity models for which the Maximum Likelihood Estimator (MLE for short) doesnot exist. We mainly illustrate the properties of our estimator on models ofdensities on the line that satisfy a shape constraint. We show that itpossesses some similar optimality properties, with regard to some global ratesof convergence, as the MLE does when it exists. It also enjoys some adaptationproperties with respect to some specific target densities in the model forwhich our estimator is proven to converge at parametric rate. More important isthe fact that our estimator is robust, not only with respect to modelmisspecification, but also to contamination, the presence of outliers among thedataset and the equidistribution assumption. This means that the estimatorperforms almost as well as if the data were i.i.d. with density $p$ in asituation where these data are only independent and most of their marginals areclose enough in total variation to a distribution with density $p$. We alsoshow that our estimator converges to the average density of the data, when thisdensity belongs to the model, even when none of the marginal densities belongsto it. Our main result on the risk of the estimator takes the form of anexponential deviation inequality which is non-asymptotic and involves explicitnumerical constants. We deduce from it several global rates of convergence,including some bounds for the minimax $\mathbb{L}_{1}$-risks over the sets ofconcave and log-concave densities. These bounds derive from some specificresults on the approximation of densities which are monotone, convex, concaveand log-concave. Such results may be of independent interest.

Robust density estimation with the $\mathbb{L}_{1}$-loss. Applications to the estimation of a density on the line satisfying a shape constraint