The Annals of Statistics

Some Incomplete and Boundedly Complete Families of Distributions

Wassily Hoeffding

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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.

Article information

Ann. Statist., Volume 5, Number 2 (1977), 278-291.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62G30: Order statistics; empirical distribution functions

Complete (incomplete) families of distributions boundedly complete families of distributions completedness relative to the permutation group invariance under permutations symmetric unbiased estimator similar tests


Hoeffding, Wassily. Some Incomplete and Boundedly Complete Families of Distributions. Ann. Statist. 5 (1977), no. 2, 278--291. doi:10.1214/aos/1176343795.

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