The narrowest confidence interval for the variance

Abstract

The construction of confidence intervals for the variance of a normal population is traditionally based on the chi-square distribution, with the total risk symmetrically allocated between the two bounds. However, due to the asymmetry of this distribution, the resulting interval is not of minimal length for a given confidence level. Although this issue has been addressed in the literature (notably by Guenther, 1969; Dahiya and Guttman, 1982), these approaches remain largely theoretical and do not always provide a simple and operational procedure for constructing the shortest confidence interval. In this paper, we propose an alternative approach based on a continuous parametrization of risk allocation and on the direct numerical optimization of the interval length under a fixed confidence level constraint. The main contribution lies in the development of a simple, reproducible, and implementable algorithm for computing the shortest confidence interval for the variance.The results show that the optimized interval is consistently shorter than the classical one. For small sample sizes (n ≤ 10), the reduction in length ranges from approximately 5% to 12%, whereas it becomes negligible for larger samples (n ≥ 50). This improvement leads to a more precise estimation of the variance without affecting the nominal coverage probability.

JEL Classification: C15
Paper type: Theoretical Research