This paper extends the empirical minimum divergence approach for models which satisfy linear constraints with respect to the probability measure of the underlying variable (moment constraints) to the case where such constraints pertain to its quantile measure (called here semi parametric quantile models). The case when these constraints describe shape conditions as handled by the L-moments is considered and both the description of these models as well as the resulting non classical minimum divergence procedures are presented. These models describe neighborhoods of classical models used mainly for their tail behavior, for example neighborhoods of Pareto or Weibull distributions, with which they may share the same first L-moments. A parallel is drawn with similar problems held in elasticity theory and in optimal transport problems. The properties of the resulting estimators are illustrated by simulated examples comparing Maximum Likelihood estimators on Pareto and Weibull models to the minimum Chi-square empirical divergence approach on semi parametric quantile models, and others.

35 pages