Robust density estimation and model selection for the L1 loss : Applications to shape-constrained density estimation.
There is a growing interest in shape-constrained methods in the field of statistical inference. The idea is to replace restrictive parametric assumptions about the target function (here a density) with a shape constraint that it must satisfy such as convexity, monotonicity, and log-concavity. The favourite estimator used in this framework is the maximum likelihood estimator (MLE) which shows good adaptation properties with respect to some specific classes of densities and reaches optimal global
In a first joint work with Y. Baraud and G. Maillard, we design, in the one-dimensional case, a general estimation procedure for the L1-loss that retains the minimax and adaptation properties of the MLE and that is also robust: it remains stable with respect to a slight deviation from an ideal situation where the data are truly i.i.d. and their density belongs to the model under consideration.
In this talk, I will present the density estimator and a model selection procedure that we are currently working on. I will illustrate both on density models where the density satisfies some shape constraint such (piecewise) monotonicity and (piecewise) convex/concavity.