Range Preserving Unbiased Estimators in the Multinomial Case
研究了基于多项分布观测估计实值函数时，无偏估计量可能不保范围的问题，给出了保范围的条件。
Abstract Consider estimating the value of a real-valued function f(p), p = (p 0, p 1, …, pr ), on the basis of an observation of the random vector X = (X 0, X 1, …, Xr ) 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 of f(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.