Unbiased Estimation of Some Multivariate Probability Densities and Related Functions

Abstract

A problem which has received considerable attention in recent years is that of estimating the tail probability in a distribution belonging to a specified family. Kolmogorov (1950), Lieberman and Resnikoff (1955), and Healy (1956) obtained the unique minimum variance unbiased (UMVU) estimator of $P\{x > c\}$, where $x$ has a normal distribution with unknown mean and variance; they used the Rao-Blackwell theorem for this purpose. Washio, Morimoto and Ikeda (1956) used integral transform theory to study the Koopman-Pitman family. Barton (1961) provided UMVU estimators of the normal, Poisson and binomial distribution functions. The most extensive study is that by Tate (1959) who obtained the UMVU estimators for the cdf (and other functionals) for several probability densities; this was accomplished by using transform theory. In the context of reliability theory, Glasser (1962) estimated the tail of the exponential distribution and Basu (1964) that of the gamma distribution; other related work in the area of reliability theory is that of Rutemiller (1966), and Zacks and Even (1966). Non-parametric estimation of probability densities has been treated by Rosenblatt (1956), Parzen (1962), Lead-better and Watson (1963), and Weiss and Wolfowitz (1967). The more general problem of estimating $Ef(x)$, where $f$ is any given function, was considered by Neyman and Scott (1960) and Schmetterer (1961) in the case that $x$ has a normal distribution with unknown mean and variance. Schmetterer's solution really consists in obtaining the UMVU estimator of the density and then integrating the product of this estimate and $f$. That this procedure is of wider applicability is seen from the following result.