Range of Correlation Matrices for Dependent Random Variables with Given Marginal Distributions
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Let X 1,...,X d be d (d≥3) dependent random variables with finite variances such that X j ∼F j . Results on the set S d (F 1,...,F d ) of possible correlation matrices with given margins are obtained; this set is relevant for simulating dependent random variables with given marginal distributions and a given correlation matrix. When F 1=...=F d =F, we let S d (F) denote the set of possible correlation matrices. Of interest is the set of F for which S d (F) is the same as the set of all non-negative definite correlation matrices; using a construction with conditional distributions, we show that this property holds only if F is a (location-scale shift of a) margin of a (d−1)-dimensional spherical distribution.
Keywords and phrasesSpherically symmetric elliptically contoured copula partial correlation Fréchet bounds
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