Some Theorems on Optimality of a Single Observation Confidence Interval for the Mean of a Normal Distribution

TL;DR

This paper investigates the optimal construction of confidence intervals for the mean of a normal distribution using a single observation, providing new minimax optimal rules and multivariate extensions.

Contribution

It introduces a specific mixture of invariant rules that achieve minimax optimality for single-observation confidence intervals in normal distributions.

Findings

Mixture of invariant rules achieves minimax optimality.

Multivariate confidence set based on a single observation vector.

Improves coverage properties over existing methods.

Abstract

We consider the problem of finding a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown. Portnoy (2017) characterizes the scale-sign invariant rules and shows that the Hunt-Stein construction provides a randomized invariant rule that improves on any given randomized rule in the sense that it has greater minimal coverage among all procedures with a fixed expected length. Mathematical results here provide a specific mixture of two non-randomized invariant rules that achieve the minimax optimality. A multivariate confidence set based on a single observation vector is also developed.