Stable Decision Problems

Abstract

A decision problem is characterized by a loss function $V$ and opinion $H$. The pair $(V, H)$ is said to be strongly stable iff for every sequence $F_n \rightarrow_\omega H, G_n \rightarrow_\omega H$ and $L_n\rightarrow V, W_n\rightarrow V$ uniformly, $\lim_{\varepsilon \downarrow 0} \lim \sup_{n\rightarrow \infty} \lbrack \int L_n(\theta, D_n(\varepsilon)) dF_n(\theta) - \inf_D \int L_n(\theta, D) dF_n(\theta)\rbrack = 0$ for every sequence $D_n(\varepsilon)$ satisfying $\int W_n(\theta, D_n(\varepsilon)) dG_n(\theta) \leqq \inf_D \int W_n(\theta, D) dG_n(\theta) + \varepsilon.$ We show that squared error loss is unstable with any opinion if the parameter space is the real line and that any bounded loss function $V(\theta, D)$ that is continuous in $\theta$ uniformly in $D$ is stable with any opinion $H$. Finally we examine the estimation or prediction case $V(\theta, D) = h(\theta - D)$, where $h$ is continuous, nondecreasing in $(0, \infty)$ and nonincreasing in $(-\infty, 0)$ and has bounded growth. While these conditions are not enough to assure strong stability, various conditions are given that are sufficient. We believe that stability offers the beginning of a Bayesian theory of robustness.