Probability-Centered Prediction Regions

Abstract

Consider the problem of constructing a prediction region $D_n$ for a potentially observable variable $X$ on the basis of a learning sample of size $n$. Usually, the requirement that $D_n$ contain $X$ with probability $\alpha$, conditionally on the learning sample, does not uniquely determine $D_n$. This paper develops a general probability-centering concept for prediction regions that extends to vector-valued or function-valued $X$ the classical notion of an equal-tailed prediction interval. The dual requirements of probability centering and specified coverage probability determine $D_n$ uniquely. Several examples illustrate the scope and consequences of the proposed centering concept.