STATISTICS & PFpBAB~~TV zyxwvuts ELSEVIER Statistics & Probability Letters 21 (1994) 395-403 Independence-distribution-preserving dependency structures for the modified likelihood ratio test for detecting unequal covariance matrices Dean M. YoungaT*, John W. Seaman Jr.a, Laurie M. Meauxb a Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Information Systems, Baylor University, P.O. Box 98005 Waco, TX 76798, USA b Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA Received October 1993 Abstract The modified likelihood ratio (MLR) test statistic is frequently used to detect unequal covariance matrices. We are concerned with examining this statistic with respect to departures from the usual i.i.d. assumptions on the sample data. In particular we characterize the joint covariance structure of two groups of multivariate normal observations so that the distribution of this MLR test statistic is identical to that under the usual assumption of independent identically distributed observations. Keywords: Correlated observations; Robustness; Wishart random matrices; Multivariate quadratic forms 1. Introduction The effect of the violation of the usual independence assumption for both parametric and nonparametric inference procedures has received much attention during the past three decades. Researchers such as Walsh (1947), Basu et al. (1974), Tubbs (1980a, b), Smith and Lewis (1980), Lawoko and McLachlan (1982), and Pavur and Davenport (1985) have shown that ignoring requisite observation dependence structures for statistics can have severe consequences. However, in some cases the sampling distribution of a statistic is not affected by non-i.i.d. observation dependence structures. That is, there are observation covariance structures for which the sampling distribution of a statistic is the same as it would be under the usual i.i.d. assumption. We shall refer to such an observation dependency structure as independence distribution preserving (IDP). The purpose of this paper is to characterize the general positive-definite IDP covariance structure for the modified likelihood ratio (MLR) test used to detect unequal covariance matrices under the multivariate normality assumption. *Corresponding author. 0167-7152/94/$7.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-7152(94)00037-9