Range Preserving Unbiased Estimators in the Multinomial Case

Consider estimating the value of a real-valued function f(p),p = (p₀,P₁. …,P_r), on the basis of an observation of the random vector X = (X₀, X₁, …,X_r) whose distribution is multinomial (n, p). It is known that an unbiased estimator exists if and only if f is a polynomial of degree at most n, in which case the unbiased estimator off(p) is unique. In general, however, this estimator has the serious fault of not being range preserving; that is, its value may fall outside the range of f(p). In this article, a condition on f is derived that is necessary for the unbiased estimator to be range preserving and that is sufficient when n is large enough.