Abstract
Consider estimating the value of a real-valued function f(p),p = (p 0,P 1. …,P r), on the basis of an observation of the random vector X = (X 0, X 1, …,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.
Wassily Hoeffding is Professor Emeritus, Department of Statistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514. The author is grateful to a referee and an associate editor for their comments, which helped improve an earlier version of this article, and to Stamatis Cambanis for his assistance in the revision.
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References
Halmos, Paul R. (1946), “The Theory of Unbiased Estimation,” Annals of Mathematical Statistics, 17, 34–43.
Hoeffding, Wassily (1983), “Unbiased Range-Preserving Estimators,” in A Festschrift for Erich L. Lehmann, eds. P. J. Bickel, K. Doksum, and J. L. Hodges, Jr., Belmont, Calif.: Wadsworth, 249–260.
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—(1959), Testing Statistical Hypotheses, New York: John Wiley.
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Hoeffding, W. (1994). Range Preserving Unbiased Estimators in the Multinomial Case. In: Fisher, N.I., Sen, P.K. (eds) The Collected Works of Wassily Hoeffding. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0865-5_43
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DOI: https://doi.org/10.1007/978-1-4612-0865-5_43
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