Abstract
An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many scientific fields including agriculture, medical, environmental and engineering applications. We discuss the property of this k-SSCS covariance structure, namely, the property of Jordan algebra. We formally show that our D2 test statistic for k-th order data is an extension or the generalization of the D2 test statistic for second-order data and for third-order data, respectively. We also derive the D2 test statistic for third-order data and illustrate its application using a medical dataset from a clinical trial study of the eye disease glaucoma. The new test statistic is very efficient for high-dimensional data where the estimation of unstructured variance-covariance matrix is not feasible due to small sample size.
Highlights
IntroductionWe study the hypotheses testing problems of equality of means for high-dimensional and higher-order (multi-dimensional arrays) data
We study the hypotheses testing problems of equality of means for high-dimensional and higher-order data
We study the tests of hypotheses of equality of means for one population as well as for two populations for high-dimensional and higher-order data with k-self similar compound symmetry (k-SSCS) covariance structure
Summary
We study the hypotheses testing problems of equality of means for high-dimensional and higher-order (multi-dimensional arrays) data. Olkin [15] studied the hypothesis testing problem of the equality of the mean vectors of multiple populations of second-order data using a 2-SSCS covariance structure, which is reminiscent of a model of Wilks [16]. Roy et al [19] and Žežula et al [1] studied the hypotheses testing problems on the mean for the second-order data using a 2-SSCS covariance structure. A majority of the above-mentioned authors only studied the second-order matrixvariate data and used a 2-SSCS covariance structure where the exchangeability (invariance) property in one factor was present. The aim of this paper is to derive a test statistic for mean for high-dimensional k-th order data using k-SSCS covariance matrix by generalizing D2 test statistics developed in Žežula et al [1].
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