Abstract
A multivariate distribution is said to have multiplicative correlation if the correlation matrix R = (r ij ) is written as r ij = δ i δ j or r ij = −δ i δ j (i ≠ j) for a parameter vector \(\mathbf{\delta}=(\delta_1,\ldots,\delta_n)\). We first determine feasible values for \(\mathbf{\delta}\) and show that variables with such a correlation matrix can always be decomposed into a common “signal” variable plus individual “noise” variables. It is also shown that a special case of this correlation matrix implies a sum constraint among variables and vice versa. Such properties illustrate why many multivariate distributions have such a correlation structure. Furthermore, several invariance properties lead to simple relations among several multivariate distributions.
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Baba, K., Shibata, R. Multiplicative Correlations. Ann Inst Stat Math 58, 311–326 (2006). https://doi.org/10.1007/s10463-006-0036-x
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DOI: https://doi.org/10.1007/s10463-006-0036-x