Misspecified Tl Tests . I . Location and Scale

Hotelling's (1931) 72 is used widely in multivariate data analysis, encompassing tests for means, the construction of confidence ellipsoids and their projections, the analysis of repeated measurements, and statistical process control using Iiotelling's (1947) 72 charts. These and other applications are valid under strict model specifications which, in practice, may fail. The knowledgeable use of 72 thus requires that its properties be understood when modcl assumptions fail. While the literature on this topic has focused largely on the failure of normality, this is but one of several critical assumptions. In this papcr we consider the misspecification of location and scale in the modcl for a multivariate experiment, noting practical circumstances in which such misspecifications may occur. To fix ideas, let Np(p, Z) be the Gaussian law on RP with mean p and dispersion matrix E; W:(',(v, E) denotes the central Wishart distribution having v degees of freedom and scale parameters C; and 2 ( Z ) signdies the law of distribution of Z . The conventional model for T2 entails a random sample ( X I , ... , X,) from Np(p, C), the unbiased sample