On Unbiased Estimation of Density Functions

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.