Gibbs Copulas

Manfred Denker () and Aleksey Min ()
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Manfred Denker: Universität Göttingen, Institut für Mathematische Stochastik
Aleksey Min: Technical University of Munich, Department of Mathematics, TUM School of Computation, Information and Technology

A chapter in Statistical Dependence Modeling, 2026, pp 53-75 from Springer

Abstract: Abstract Real valued stationary processes with arbitrary marginal distributions can be factorized by stationary processes with uniform marginals and the same probabilistic mixing properties. Here we show that this phenomena persists with “copulas” for certain processes: Given a d-dimensional process $$X_n=(X_n^1,...,X_n^d)$$ X n = ( X n 1 , . . . , X n d ) , $$n\ge 1$$ n ≥ 1 , which has a Gibbs distribution for some potential $$\varphi $$ φ , we show that there exists a unique potential $$\tilde{\varphi }$$ φ ~ and a d-dimensional stationary random process $$U_n=(U_n^1,...,U_n^d)$$ U n = ( U n 1 , . . . , U n d ) ( $$n\ge 1$$ n ≥ 1 ), with uniform marginals for each $$U_n^i$$ U n i which represents a Gibbs distribution for the potential $$\tilde{\varphi }$$ φ ~ and which has the same copulas as the original process. In case the potential is given by a linear model $$\varphi _0+\sum _{i=1}^r \vartheta _i \varphi _i$$ φ 0 + ∑ i = 1 r ϑ i φ i ( $$\vartheta =(\vartheta _i)_{1\le i\le r}$$ ϑ = ( ϑ i ) 1 ≤ i ≤ r ) we also show that the maximum likelihood estimator for the parameter $$\vartheta $$ ϑ exists and determine its asymptotic distribution.

Keywords: Copula; Gibbs distribution; Maximum likelihood estimator; 62F10; 62F25; 62M09 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-14252-8_4

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DOI: 10.1007/978-3-032-14252-8_4

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