Statistical Modelling 13 (5&6) (2013), 431–457
Finite-sample inference with monotone incomplete multivariate normal data, III: Hotelling’s T2-statistic
Megan M Romer
Department of Statistics,
Pennsylvania State University,
University Park,
PA 16802
Donald St. P Richards,
Department of Statistics,
Pennsylvania State University,
University Park,
PA 16802
USA
e-mail: richards@stat.psu.edu
Abstract:
In the setting of inference with two-step monotone incomplete data drawn from Nd (µ, Σ), 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.
Keywords:
asymptotic distribution; confidence region; equivariance; Hotelling’s T2-squared statistic; maximum likelihood estimator; missing completely at random; missing data; monotone incomplete data; multivariate normal distribution; orthogonal invariance; pivotal quantity; stochastic representation; simultaneous confidence intervals
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