On the Estimation of Mixing Distributions

Abstract

Let $\mathscr{F} = \{F(x; y), y \varepsilon E\}$ be a family of cumulative distribution functions (cdf's) in the variable $x$ indexed by $y \varepsilon E$, where $E$ is a measurable subset of the real line. Assume that $F(x; y)$ is measurable in $y$ for all $x$. Then, for any nondegenerate cdf $G$, whose induced probability measure assigns measure one to $E$, the cdf $H_G(x) = \int_E F(x; y) dG(y)$ is called a "mixture" of $\mathscr{F}$, and $G$ is called the "mixing distribution." The family $\mathscr{F}$ will be called the kernel of the mixture. Among the many interesting problems engendered by mixtures of distributions is one of great practical importance, namely, the estimation of the mixing distribution. This problem is simply stated--on the basis of observations from the mixture $H_G$, estimate the mixing distribution $G$. However, before actual estimation can be meaningfully investigated, identifiability of the family of mixtures, defined presently, must be verified. Let $\mathscr{G}$ denote the class of mixing distributions $G$, and $\mathscr{H}$ the induced class of mixtures $H$ for some specified family $\mathscr{F}$. A mixture of $\mathscr{F}$ will be called "identifiable" if $H(x) = \int F(x; y) dG^\ast(y) = \int F(x; y) dG(y)$ implies $G^\ast = G$. If every member $H$ of $\mathscr{H}$ is identifiable, then $\mathscr{H}$ will be said to be identifiable. For identifiable families $\mathscr{H}$, the problem of estimation of the mixing cdf when the elements of $\mathscr{F}$ occurring in the mixture are known is dealt with here. In particular, the problem of unbiased estimation for finite mixtures is considered. In a finite mixture the kernel is any finite set of known but arbitrary cdf's and the mixing distributions are discrete assigning positive weight to each of the cdf's in the kernel. The estimation of the mixing ratio when two arbitrary known cdf's are mixed is discussed at length in Section 2. Identifiability is then evident. Necessary and sufficient conditions on two distribution mixtures for uniform attainment of the Cramer-Rao lower bound are derived. The class of $\theta$-efficient estimators is found; also, as it turns out, the minimax unbiased estimator is a member of this $\theta$-efficient class and it is characterized. In Section 3, the results of Section 2 are extended to any finite mixture. For identifiable finite mixtures, necessary and sufficient conditions for the existence of an estimator which uniformly attains the minimal ellipsoid of concentration are given. The $\theta$-efficient family of estimators is derived; also, estimators within the $\theta$-efficient family which are consistent asymptotically normal efficient are characterized.