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On Non‐Normal Invariance Principles for Multi‐Response Permutation Procedures
Author(s) -
Brockwell Peter J.,
Mielke Paul W.,
Robinson John
Publication year - 1982
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1982.tb00805.x
Subject(s) - mathematics , invariant (physics) , asymptotic distribution , invariance principle , permutation (music) , univariate , distribution (mathematics) , statistic , test statistic , euclidean distance , chi square test , resampling , combinatorics , f distribution , mathematical analysis , statistics , statistical hypothesis testing , probability distribution , multivariate statistics , physics , mathematical physics , geometry , linguistics , philosophy , estimator , acoustics
Summary A non‐normal invariance principle is established for a restricted class of univariate multi‐response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi‐squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d , and it is shown that if d (x, y) = |x — y| u , u† 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).