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.

翻译：在有限空间上有限测度的情形下，我们给出了$φ$-投影连续可微的条件。当投影目标集合为凸集时，我们证明了所需假设可由易于验证的条件导出。特别地，对于输入概率向量及一类相当广泛的φ散度，我们得到在线性等式定义的集合上进行投影时$φ$-投影具有连续可微性。所得结果被应用于推导$φ$-投影估计量（即最小φ散度估计量）的渐近性质，包括在参数化概率向量集合、具有固定矩的分布生成的概率向量集合以及二元概率阵列的Fr\'echet类上的投影情形。无论待投影元素是否属于目标投影集合，所得渐近结论均成立。