On Non‐Normal Invariance Principles for Multi‐Response Permutation Procedures
Peter J. Brockwell, Paul W. Mielke, John Robinson · Australian Journal of Statistics · 1982
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).