A characterization of the independence - distribution - preserving covariance structure for the multivariate maximum squared - radii statistic

Necessary and sufficient conditions on the observation covariance structure and on the set of linear transformations are given for which the distribution of the multivariate maximum squared - radii statistic for detecting a single multivariate outlier is invariant from the distribution assuming the usual independence covariance structure. Thus, we extend the work of Baksalary and Puntanen (1990), who have given necessary and sufficient conditions for an independence-distribution-preserving covariance structure for Grubbs' statistic for detecting a univariate outlier. We also extend the work of Marco, Young, and Turner (1987) and Pavur and Young (1991), who have given sufficient conditions for an independence-distribution-preserving dependency structure for the multivariate squared - radii statistic.     

INDEPENDENCE DISTRIBUTION-PRESERVING NONNEGATIVE-DEFINITE COVARIANCE STRUCTURES FOR THE SAMPLE VARIANCE

This paper explicitly characterizes the general nonnegative-definite covariance structure for a multivariate normal random vector such that the univariate sample variance is distributed exactly as if the sample observations are normal independent and identically distributed (i.i.d.). This work extends the results of Baldessari (1965) and Stadje (1984) who have characterized the general positive-definite covariance matrix such that the univariate sample variance is distributed exactly as if the sample observations are normal i.i.d.     

A characterization of nonnegative-definite independence distribution-preserving covariance structures for the maximum squared-radii statistic

The multivariate maximum squared-radii (MMSR) statistic is commonly used to detect multivariate outliers. We characterize the general form of the nonnegative-definite observation covariance structure for which the distribution of the MMSR statistic is the sameas the distribution resulting from the usual independence covariance structure. Thus, we extend the work of Young, Seaman, and Meaux (1992), who have characterized the general form of the positive-definite independence-distribution-preserving (IDP) dependency structure for the MMSR statistic. We also improve upon the results of Younget al (1992) in that we give a more complete and simple proof of the characterization of the general positive-definite IDP covariance structure for the MMSR statistic.