Small-sample confidence interval estimation of the common mean value of a multivariate normal distribution

Abstract

We derive two new interval estimators for the common mean of a multivariate normal distribution, the general-t confidence interval, and an integrated-likelihood-ratio (ILR) confidence interval. Our numerical evaluations, Monte Carlo simulations, and two real-data-example results suggest that for many realistic multivariate covariance matrices, our general-t interval yields more precise confidence intervals than the conditional-t or ILR confidence intervals when the sample size is small relative to the number of parameters to be estimated. We also prove that for a general class of covariance structures, the general-t interval yields narrower expected lengths than the conditional-t interval proposed by Halperin (1961) for all samples of size two or more. Additionally, via a Monte Carlo simulation, we demonstrate that for a fixed sample size, a confidence interval studied in Krishnamoorthy and Lu (2005) consisting of the shortest of the computed univariate marginal-t intervals yields sub-nominal coverage that becomes increasingly sub-nominal as the multivariate data dimension increases.