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Inference in ϕ-families of distributions

This paper is devoted to the study of the parametric family of multivariate distributions obtained by minimizing a convex functional under linear constraints. Under certain assumptions on the convex functional, it is established that this family admits an affine parameterization, and parametric estimation from an i.i.d. random sample is studied. It is also shown that the members of this family are the limit distributions arising in inference based on empirical likelihood. As a consequence, given a probability measure μ0 and an i.i.d. random sample drawn from μ0, nonparametric confidence domains on the generalized moments of μ0 are obtained.