Finite-sample inference with monotone incomplete multivariate normal data, III: Hotelling’s T2-statistic

Abstract

In the setting of inference with two-step monotone incomplete data drawn from Nd( µ, ∑), a multivariate normal population with mean µ and covariance matrix ∑, we derive a stochastic representation for the exact distribution of a generalization of Hotelling’s T2-statistic, thereby enabling the construction of exact level ellipsoidal confidence regions for µ. By applying the equivariance of [Formula: see text] and [Formula: see text] the maximum likelihood estimators of µ and ∑, respectively, we show that the T2-statistic is invariant under affine transformations. Further, as a consequence of the exact stochastic representation, we derive upper and lower bounds for the cumulative distribution function of the T2-statistic. We apply these results to construct simultaneous confidence regions for linear combinations of µ, and we apply these results to analyze a dataset consisting of cholesterol measurements on a group of Pennsylvania heart disease patients.