On the differentiability of $\phi$-projections in the discrete finite case

In the case of finite measures on finite spaces, we state conditions under which $\phi$-projections are continuously differentiable. When the set on which one wishes to $\phi$-project is convex, we show that the required assumptions are implied by easily verifiable conditions. In particular, for input probability vectors and a rather large class of \phidivergences, we obtain that $\phi$-projections are continuously differentiable when projecting on a set defined by linear equalities. The obtained results are applied to the derivation of the asymptotics of $\phi$-projection estimators (that is, minimum \phidivergence\, estimators) when projecting on parametric sets of probability vectors, on sets of probability vectors generated from distributions with certain moments fixed and on Fr\'echet classes of bivariate probability arrays. The resulting asymptotics hold whether the element to be $\phi$-projected belongs to the set on which one wishes to $\phi$-project or not.