Improved Invariant Confidence Intervals for a Normal Variance

Abstract

Confidence intervals for the variance of a normal distribution with unknown mean are constructed which improve upon the usual shortest interval based on the sample variance alone. These intervals have guaranteed coverage probability uniformly greater than a predetermined value $1-\alpha$ and have uniformly shorter length. Using information relating the size of the samples mean to that of the sample variance, we smoothly shift the usual minimum length interval closer to zero, simultaneously bringing the endpoints closer to each other. The gains in coverage probability and expected length are also investigated numerically. Lastly, we examine the posterior probabilities of the intervals, quantities which can be used as post-data confidence reports.