Abstract
Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones.
Highlights
Canonical correlation analysis is a multivariate statistical method purported to analyze the correlation structure between two random vectorsCitation: Loperfido, N.M.R
This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where Canonical correlation analysis (CCA) provides an insight into their nonlinear dependence structure
Perturbed independence distributions are closely related to skew-symmetric ones
Summary
The following theorem shows that the canonical covariates belonging to different canonical vectors and with different indices are independent, if the joint distribution of the original variables is perturbed independence with sign-symmetric components. The following corollary of the above theorem shows that the same property still holds true when the original variables have a perturbed independence distribution with normal components. If the joint distribution of x and y is a perturbed independence model with components h(·) and k(·), location vectors μ and ν, perturbing function π(·) and association matrix Ψ we have fxy(t, s; μ, ν,Ψ) = 2h(t − μ)k(s − ν)π (s − ν) Ψ(t − μ) , fx(t;μ) = h(t − μ), fy(s;ν) = k(s − ν), π(−a) = 1 − π(a).
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