## The Annals of Mathematical Statistics

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

### On Unbiased Estimation of Density Functions

A. H. Seheult and C. P. Quesenberry

#### 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