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Finite-sample inference with monotone incomplete multivariate normal data. I. (English) Zbl 1171.62038

Summary: We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from \(N_d(\mu,\Sigma)\), a multivariate normal population with mean \(\mu\) and covariance matrix \(\Sigma\). We derive a stochastic representation for the exact distribution of \(\widehat\mu\), the maximum likelihood estimator of \(\mu\). We obtain ellipsoidal confidence regions for \(\mu\) through \(T^2\), a generalization of Hotelling’s statistic. We derive the asymptotic distribution of, and probability inequalities for, \(T^2\) under various assumptions on the sizes of the complete and incomplete samples. Further, we establish an upper bound for the supremum distance between the probability density functions of \(\widehat\mu\) and \(\widetilde\mu\), a normal approximation to \(\widehat\mu\).

MSC:

62H12 Estimation in multivariate analysis
62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
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