Unbiased Estimation in Convex Families

Abstract

Suppose that we observe random variables $X_1, \cdots, X_n$ which are identically and independently distributed according to some distribution $F$ where $F$ ranges over a family $\mathscr{F}$. The following question was posed by J. Steffensen in this abstract context in [7]. (1) If a functional $q$ is given on $\mathscr{F}$, when does there exist a statistic $\delta(X_1, \cdots, X_n)$ such that, \begin{equation*} \tag{1.1} E_F(\delta(X_1, \cdots, X_n)) = q(F),\end{equation*} for all $F \varepsilon \mathscr{F}$? (As usual $E_F$ denotes the expectation under the assumption that $F$ is the common distribution of the $X_i$.) In other words, when does there exist an unbiased estimate of $q(F)$ based on $n$ observations? A question which naturally follows from question 1 was raised and considered by Halmos in [2]. (2) Let the degree of $q$ be the smallest $n \leqq \infty$ such that (1.1) holds for all $F \varepsilon \mathscr{F}$ and some $\delta$. Characterize the degree of estimable functionals $q$ for various families $\mathscr{F}$. (By estimable we mean merely that (1.1) should hold for some finite $n$ and $\delta$.) In his paper [2] Halmos dealt with question 2 as well as the more important issue of existence of best unbiased estimates in the context of families $\mathscr{F}$ which are very large, and asked whether anything can be said about more interesting families $\mathscr{F}$. The next two sections of our paper deal exclusively with question 2 for families $\mathscr{F}$ which are closed under finite mixtures (convex combinations). In Section 2, some examples lead to our main result, Theorem 2.1, which gives an elementary characterization of degree for functionals defined on families $\mathscr{F}$ as above. The theorem is followed by further examples which illustrate its utility. (However, the required convexity property excludes the usual parametric families.) Theorem 2.2, an extension of a theorem of Halmos [2], characterizes the degree of a product of estimable functionals. It is applied to give the degree of the $r$th cumulant of a distribution when $\mathscr{F}$ is the family of all distributions with moments of all orders. The third section deals with the problem of degree when we observe samples from two or more populations with different distributions. Theorem 3.1 shows that the notion of degree can be successfully extended in this situation. The theorem is followed by several examples in which the degree of various functionals such as two-sample distance criteria is computed. Finally, in Section 4 we address ourselves to question 1. Theorem 4.1 gives a simple characterization of estimable functionals when the family $\mathscr{F}$ is $\mathscr{F}(\mu)$ where, \begin{equation*} \tag{1.2} \mathscr{F}(\mu) = \{F:F \text{is absolutely continuous with respect to} \mu\}\end{equation*} for a fixed $\sigma$ finite measure $\mu$. The characterization is suggested by Theorem 2.1. After some examples we state without proof a further theorem (4.2) along these lines and illustrate its applicability.