Some Incomplete and Boundedly Complete Families of Distributions

Abstract

Let $\mathscr{P}$ be a family of distributions on a measurable space such that $(\dagger) \int u_i dP = c_i, i = 1, \cdots, k$, for all $P\in\mathscr{P}$, and which is sufficiently rich; for example, $\mathscr{P}$ consists of all distributions dominated by a $\sigma$-finite measure and satisfying $(\dagger)$. It is known that when conditions $(\dagger)$ are not present, no nontrivial symmetric unbiased estimator of zero (s.u.e.z.) based on a random sample of any size $n$ exists. Here it is shown that (I) if $g(x_1, \cdots, x_n)$ is a s.u.e.z. then there exist symmetric functions $h_i(x_1, \cdots, x_{n - 1}), i = 1, \cdots, k$, such that $g(x_1, \cdots, x_n) = \sum^k_{i = 1} \sum^n_{j = 1} \{u_i(x_j) - c_i\}h_i(x_1, \cdots, x_{j - 1}, x_{j + 1}, \cdots, x_n);$ and (II) if every nontrivial linear combination of $u_1, \cdots, u_k$ is unbounded then no bounded nontrivial s.u.e.z. exists. Applications to unbiased estimation and similar tests are discussed.