The joint distribution of Studentized residuals under elliptical distributions

Scaled and Studentized statistics are encountered frequently, and they often play a decisive role in statistical inference and testing. For instance, taking the sample mean vector X̄=∑j=1NXj/N and the sample covariance matrix S=∑j=1N(Xj−X̄)(Xj−X̄)′/(N−1) for an iid sample {Xj}j=1N, some statistics f...

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Bibliographic Details
Published in:Journal of Multivariate Analysis Vol. 128; pp. 203 - 209
Main Authors: Iwashita, Toshiya, Klar, Bernhard
Format: Journal Article
Language:English
Japanese
Published: New York Elsevier Inc 01.07.2014
Elsevier BV
Taylor & Francis LLC
Subjects:
ddc:510
Elliptical distribution
info:eu-repo/classification/ddc/510
Left-spherical distribution
Mathematics
Matrix
Normal distribution
Residuals
Scaled residuals
Spherical distribution
Statistical inference
Statistics
Studentized statistic
Studies
Vector space
secondary
Spherical distribution
Studentized statistic
Left-spherical distribution
Scaled residuals
Elliptical distribution
primary
ISSN:0047-259X, 1095-7243
Online Access:Get full text
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Summary:Scaled and Studentized statistics are encountered frequently, and they often play a decisive role in statistical inference and testing. For instance, taking the sample mean vector X̄=∑j=1NXj/N and the sample covariance matrix S=∑j=1N(Xj−X̄)(Xj−X̄)′/(N−1) for an iid sample {Xj}j=1N, some statistics for testing normality of the underlying distribution consist of the scaled residuals (the Studentized residuals or the transformed samples), Yj=S−1/2(Xj−X̄)(j=1,2,…,N). In this paper, the distribution of the random matrix the columns of which consist of the scaled residuals is derived under elliptical distributions. Also exact distributions of Studentized statistics are discussed as an application of the main result.
Bibliography:SourceType-Scholarly Journals-1
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content type line 14
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2014.03.015