We consider the minimization problem of $phi$-divergences between a given probability measure $P$ and subsets $Omega$ of the vector space $mathcal{M}_mathcal{F}$ of all signed finite measures which integrate a given class $mathcal{F}$ of bounded or u
nbounded measurable functions. The vector space $mathcal{M}_mathcal{F}$ is endowed with the weak topology induced by the class $mathcal{F}cup mathcal{B}_b$ where $mathcal{B}_b$ is the class of all bounded measurable functions. We treat the problems of existence and characterization of the $phi$-projections of $P$ on $Omega$. We consider also the dual equality and the dual attainment problems when $Omega$ is defined by linear constraints.
We introduce estimation and test procedures through divergence minimiza- tion for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized emp
irical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alterna- tives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.
We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composit
e hypotheses, extending maximum likelihood technique. An other view at the maximum likelihood approach, for estimation and test, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given both under the null and the alternative hypotheses, and approximation of the power functions is deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on $chi ^{2}$-divergence on signed finite measures and duality technique.
We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. Several statistical examples and motivations are given. These procedures extend the empirical likelihood (EL)
method and share common features with generalized empirical likelihood (GEL). We treat the problems of existence and characterization of the divergence projections of probability measures on sets of signed finite measures. Our approach allows for a study of the estimates under misspecification. The asymptotic behavior of the proposed estimates are studied using the dual representation of the divergences and the explicit forms of the divergence projections. We discuss the problem of the choice of the divergence under various respects. Also we handle efficiency and robustness properties of minimum divergence estimates. A simulation study shows that the Hellinger divergence enjoys good efficiency and robustness properties.
