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

In the setting of inference with two-step monotone incomplete data drawn from N_d(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 μ̂ and Σ̂, 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.