UNBIASED ESTIMATORS IN THE MULTINOMIAL CASE

It is known that an unbiased estimator of f(p) for binomial B(n,p) exists if and only if f is a polynomial of degree at most n, in which case the unbiased estimator of a real-valued function $f(p), p = (p_0,p_1,\cdots,p_r)$ is unique. In general, 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 a real-valued function f is derived that is necessary for the unbiased estimator to be range preserving that this is sufficient when n is large enough.