Exact probabilities of correct classifications for uncorrelated repeated measurements from elliptically contoured distributions

D. Krause and W. -D. Richter

Journal of Multivariate Analysis, 2004, vol. 89, issue 1, 36-69

Abstract: Euclidean distance-based classification rules are derived within a certain nonclassical linear model approach and applied to elliptically contoured samples having a density generating function g. Then a geometric measure theoretical method to evaluate exact probabilities of correct classification for multivariate uncorrelated feature vectors is developed. When doing this one has to measure suitably defined sets with certain standardized measures. The geometric key point is that the intersection percentage functions of the areas under investigation coincide with those of certain parabolic cylinder type sets. The intersection percentage functions of the latter sets can be described as threefold integrals. It turns out that these intersection percentage functions yield simultaneously geometric representation formulae for the doubly noncentral g-generalized F-distributions. Hence, we get beyond new formulae for evaluating probabilities of correct classification new geometric representation formulae for the doubly noncentral g-generalized F-distributions. A numerical study concerning several aspects of evaluating both probabilities of correct classification and values of the doubly noncentral g-generalized F-distributions demonstrates the advantageous computational properties of the present new approach. This impression will be supported by comparison with the literature. It is shown that probabilities of correct classification depend on the parameters of the underlying sample distribution through a certain well-defined set of secondary parameters. If the underlying parameters are unknown, we propose to estimate probabilities of correct classification.

Keywords: Classification; probabilities; Allocation; Uncorrelated; repeated; measurements; Elliptically; contoured; samples; Exact; distributions; Doubly; noncentral; generalized; F-distribution; Intersection; percentage; function; Geometric; representation; formulae (search for similar items in EconPapers)
Date: 2004
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