The Annals of Mathematical Statistics

On Unbiased Estimation of Density Functions

A. H. Seheult and C. P. Quesenberry

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Abstract

Let $X^{(n)} = (X_1, \cdots, X_n)$ be a random sample of size $n$ from the distribution of a real-valued random variable $X$ with an absolutely continuous distribution function $F$ and a density function $f$. Rosenblatt (1956) showed that in this setting there exists no unbiased estimator of $f$ based on the order statistics. His result follows from the fact that the empirical distribution function is not absolutely continuous. He also assumed that $f$ is continuous, but this condition is unnecessary. Rosenblatt's result also arises as a consequence of general results by Bickel and Lehmann (1969) on unbiased estimation in convex families, such as the family of all such $F$ (above). A number of writers (Kolmogorov (1950), Schmetterer (1960), Ghurye and Olkin (1969)) have obtained unbiased estimators of particular normal-related families as well as for other estimable functions. Washio, Morimoto and Ikeda (1956) considered related questions for the Koopman-Pitman family of densities, and Tate (1959) confined his attention to functions of scale and location parameters. A question arises as to exactly when unbiased--uniform minimum variance unbiased (UMVU)--estimators of density functions exist and when they do not. In a recent publication, Lumel'skii and Sapozhnikov (1969) considered such a question in relation to estimating the density function at a point, whereas, in this paper our definition of unbiasedness requires the estimator to be unbiased at every point. The so-called "Bayesian" methods they employ yield estimators for most of the well-known families of distributions as well as for several types of $p$-dimensional discrete distributions. In Section 2 we formulate the problem in a fairly general setting and obtain results in terms of unbiased estimators of probability measures (or distribution functions) which always exist. In Section 3 we consider examples to illustrate the theory of the preceding section and in Section 4 give a theorem which generalizes a lemma stated by Ghurye and Olkin (1969) which formalizes the approach used by Schmetterer (1960) for obtaining unbiased estimators of certain types of parametric functions.

Article information

Source
Ann. Math. Statist., Volume 42, Number 4 (1971), 1434-1438.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693255

Digital Object Identifier
doi:10.1214/aoms/1177693255

Zentralblatt MATH identifier
0249.62038

JSTOR
links.jstor.org

Citation

Seheult, A. H.; Quesenberry, C. P. On Unbiased Estimation of Density Functions. Ann. Math. Statist. 42 (1971), no. 4, 1434--1438. doi:10.1214/aoms/1177693255. https://projecteuclid.org/euclid.aoms/1177693255


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