Characterization of distributions symmetric with respect to a group of transformations and testing of corresponding statistical hypothesis

It is shown that for a real orthogonal matrix A, a real number r∈(0,2), and two i.i.d. random vectors X and Y, the inequality E||X−AY|| r⩾ E||X−Y|| r is valid, with equality if and only if the distribution of X is invariant with respect to the group generated by the matrix A. Some generalizations of...

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Bibliographic Details
Published in:Statistics & probability letters 2001-06, Vol.53 (3), p.241-247
Main Authors: Klebanov, L.B., Kozubowski, T.J., Rachev, S.T., Volkovich, V.E.
Format: Article
Language:English
Subjects:
Empirical characteristic function
Empirical characteristic function Moment properties Negative-definite kernel Probability metric Stochastic inequality
Moment properties
Negative-definite kernel
Probability metric
Stochastic inequality
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Summary:It is shown that for a real orthogonal matrix A, a real number r∈(0,2), and two i.i.d. random vectors X and Y, the inequality E||X−AY|| r⩾ E||X−Y|| r is valid, with equality if and only if the distribution of X is invariant with respect to the group generated by the matrix A. Some generalizations of this property are also given and a statistical test for the corresponding hypothesis is proposed.
ISSN:0167-7152
1879-2103
DOI:10.1016/S0167-7152(01)00011-6