Finite-sample inference with monotone incomplete multivariate normal data,III: Hotelling’s T 2 -statistic

Abstract

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 $$\widehat{\mu }$$ and $$\widehat{\Sigma },$$ 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.