Title: Projection Theorems of Divergences and Likelihood Maximization Methods,
Abstract: Projection theorems of divergences enable us to find reverse projection of a
divergence on a specific statistical model as a forward projection of the
divergence on a different but rather "simpler" statistical model, which, in
turn, results in solving a system of linear equations. Reverse projection of
divergences are closely related to various estimation methods such as the
maximum likelihood estimation or its variants in robust statistics. We consider
projection theorems of three parametric families of divergences that are widely
used in robust statistics, namely the Rényi divergences (or the Cressie-Reed
power divergences), density power divergences, and the relative
$\alpha$-entropy (or the logarithmic density power divergences). We explore
these projection theorems from the usual likelihood maximization approach and
from the principle of sufficiency. In particular, we show the equivalence of
solving the estimation problems by the projection theorems of the respective
divergences and by directly solving the corresponding estimating equations. We
also derive the projection theorem for the density power divergences.