A Confidence Interval for the Center of an Unknown Unimodal Distribution Based on a Sample of Size 1

Contrary to popular belief, it is sometimes possible to produce a confidence interval for the center (e.g., mode) of a distribution based on a sample of size 1 without having any previous knowledge of the degree of its dispersion.This (possibly counterintuitive) fact was first proved by Abbot and Rosenblatt, with the first application being developed by Robert Machol, for the case of a normal variable with unknown variance.Although he and others have subsequently generalized these results, they have previously appeared only in the engineering literature, so statisticians may not be aware of them.It is the purpose of this article to present a simple proof of the validity of the confidence interval procedure for the mode of an unknown unimodal density (i.e., a density that is decreasing as its argument moves away from the mode) based on a single observation, with a comment about similar procedures for the center of a symmetric density and that of a normal density (both mean and variance being unknown).