Tests and estimation strategies associated to some loss functions

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