Extreme change-point detection

Abstract

We examine rules for predicting whether a point in $\mathbb{R}$ generated from a 50-50 mixture of two different probability distributions came from one distribution or the other, given limited (or no) information on the two distributions, and, as clues, one point generated randomly from each of the two distributions. We prove that nearest-neighbor prediction does better than chance when we know the two distributions are Gaussian densities without knowing their parameter values. We conjecture that this result holds for general probability distributions and, furthermore, that the nearest-neighbor rule is optimal in this setting, i.e., no other rule can do better than it if we do not know the distributions or do not know their parameters, or both.