Divergences and duality for estimation and test under moment condition model

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood (GEL) approach. We treat the problems of existence and characterization of the divergence projections of probability measures on sets of signed finite measures. Our approach allows to obtain the limit distributions of the estimates and test statistics (including the EL ones) under alternatives and misspecification. The asymptotic behavior of the estimates and test statistics are studied both under the model and under alternatives including misspecification, using the dual representation of the divergences and the explicit forms of the divergence projections. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.