  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 composite 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.
