Upper and Lower Posterior Probabilities for Truncated Means

Abstract

In the recent literature a system of inference which leads to upper and lower posterior distributions based on sample data has been proposed by Dempster (1966, 1967, 1968). The purpose of this paper is to apply this method to the problem of finding the upper and lower probabilities that the mean of a distribution falls within a given interval, the only prior information being that the distribution is continuous. Included in the class of all continuous distributions are those having the property that $0 < F(y) < 1$ for all real $y$. For distributions of this type the approach used here leads to trivial results. The reason for this is discussed in Section 2. To circumvent this difficulty the mean of the truncated distribution is used. That is, for given $\varepsilon_1$ and $\varepsilon_2$ such that $0 < \varepsilon_1 < 1 - \varepsilon_2 < 1$, \begin{equation*}\tag{1.1} \mu(\varepsilon_1, \varepsilon_2) = \frac{1}{1 - \varepsilon_1 - \varepsilon_2} \int^{\xi 2}_{\xi 1} td F(t)\end{equation*} where $F$ is a distribution function and $\xi_1 < \xi_2$ are real constants such that $F(\xi_1) = \varepsilon_1$ while $F(\xi_2) = 1 - \varepsilon_2$. The device of truncating the mean is to some extent artificial and is unnecessary when the continuous distribution function is constant except on an interval of finite length. However, it does offer an approach to the more general (and in the author's opinion more interesting) problem where no restriction is placed on the interval on which the distribution function is neither zero nor one. Section 2 contains a brief restatement of some of Dempster's basic definitions in the context of this particular problem along with an exposition of the type of reasoning required whenever this method is applied to problems involving truncated moments. Exact expressions for the upper and lower probabilities are derived in Section 3, and some asymptotic results are presented in Section 4. In particular, it is shown that if the upper and lower probabilities converge, they converge to the same limit. All relevant distribution theory is presented in an appendix.