  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.
