%0 Journal Article
%T Exact Inference for the Dispersion Matrix
%A Alan D. Hutson
%A Gregory E. Wilding
%A Jihnhee Yu
%A Albert Vexler
%J Advances in Statistics
%D 2014
%R 10.1155/2014/432805
%X We develop a new and novel exact permutation test for prespecified correlation structures such as compound symmetry or spherical structures under standard assumptions. The key feature of the work contained in this note is the distribution free aspect of our procedures that frees us from the standard and sometimes unrealistic multivariate normality constraint commonly needed for other methods. 1. Introduction Let be an iid -dimensional multivariate sample from an absolutely continuous distribution with dispersion matrix of as Inference about the dispersion matrix takes the general form where we assume that is specified in a particular manner, for example, a block diagonal matrix or a spherical type structure or simply an unstructured form. In general, research and testing methods of this form assume an underlying multivariate normal distribution with associated exact and approximate tests; for example, for a thorough overview and history of this testing problem, see Seber [1] and the references therewithin. In practice one can safely say that it would be rare that the multivariate normality assumption holds. Hence, we were motivated to develop an exact permutation method approach to this problem. To the best of our knowledge no so-called exact permutation tests have been developed or explored with the exception of the very special case of dimensions and testing ; for example, see Good [2]. Martin [3] provides a bootstrap algorithm for testing , which asymptotically can be shown to have the appropriate type I error rate. Unfortunately, the bootstrap methods given by Martin [3] relative to first standardizing the variables and rotating the data so as to transform the problem to the setting of testing do not work in the permutation setting. The permutation test for the case follows by permuting the second column of the data matrix and calculating the test statistic , where refers to the standard sample Pearson correlation coefficient, over all permutations. This can be done directly via a computationally expensive algorithm or via the more widely used Monte Carlo techniques. With respect to the Monte Carlo methods we generate random permutations of the data and denote the permuted value of the test statistic by . Then the one-sided value for the alternative is given as , where the index corresponds to a given permutation and denotes the indicator function. Alternative approaches found in software packages such as SAS PROC FREQ (SAS version 9.3, Cary, NC) utilize hypergeometric probabilities similar to how FisherĄ¯s exact test is carried out via treating the
%U http://www.hindawi.com/journals/as/2014/432805/