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In the parametric development of statistical inference it often is assumed that observations are independent and Gaussian. The Gaussian assumption sometimes is justified on appeal to central limit theory or on the grounds that certain normal theory procedures are robust. The independence assumption, usually unjustified, routinely facilitates the derivation of needed distribution theory.
In this thesis a variety of standard tests for scale parameters is considered when the observations are not necessarily either Gaussian or independent. The distributions considered are the spherically symmetric vector laws, i.e. laws for which x(nx1) and Px have the same distribution for every (nxn) orthogonal matrix P, and natural extensions of these to laws of random matrices. If x has a spherical law, then the distribution of Ax + b is said to be elliptically symmetric.
The class of spherically symmetric laws contains such heavy-tailed distributions as the spherical Cauchy law and other symmetric stable distributions. As such laws need not have moments, the emphasis here is on tests for scale parameters which become tests regarding dispersion parameters whenever second-order moments are defined.
Using the principle of invariance it is possible to characterize the invariant tests for certain hypotheses for all elliptically symmetric distributions. The particular problems treated are tests for the equality of k scale parameters, tests for the equality of k scale matrices, tests for sphericity, tests for block diagonal structure, tests for the uncorrelatedness of two variables within a set of m variables, and tests for the hypothesis of equi-correlatedness. In all cases except the last three the null and non-null distributions of invariant statistics are shown to be unique for all elliptically symmetric laws. The usual normal-theory procedures associated with these particular testing problems thus are exactly robust, and many of their known properties extend directly to this larger class.
In the last three cases, the null distributions of certain invariant statistics are unique but the non-null distributions depend on the underlying elliptically symmetric law. In testing for block diagonal structure in the case of two blocks, a monotone power property is established for the subclass of all elliptically symmetric unimodal distributions. |
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