Title: Some Theorems on Optimality of a Single Observation Confidence Interval for the Mean of a Normal Distribution,
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.