PreprintCC BY 4.0arXiv
Copula Operad and Copula Entropy
Abstract
We construct a symmetric operad C on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass Cac---which coincides with the L1 class of copula densities---forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula g(v)=ϕ(Ψ1(v(1)),…,Ψn(v(n)))∏k=1nψk(v(k)). Furthermore, we show that copulas with finite copula entropy---identified with the Llog L class of copula densities---are closed under substitution and hence constitute a suboperad CLlog L. On this suboperad, copula entropy is strictly additive: H(γ(Φ;Ψ1,…,Ψn))=H(Φ)+∑k=1nH(Ψk).
Related
BibTeX
@misc{lu2026copula,
title = {{Copula Operad and Copula Entropy}},
author = {Xuexing Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20512},
eprint = {2609.20512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20512},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Copula Operad and Copula Entropy
AU - Xuexing Lu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.20512
UR - https://arxiv.org/abs/2609.20512
AB - We construct a symmetric operad C on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass C^ac---which coincides with the L^1 class of copula densities---forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula g(v)=ϕ(Ψ_1(v^(1)),…,Ψ_n(v^(n)))∏_k=1^nψ_k(v^(k)). Furthermore, we show that copulas with finite copula entropy---identified with the Llog L class of copula densities---are closed under substitution and hence constitute a suboperad C^Llog L. On this suboperad, copula entropy is strictly additive: H(γ(Φ;Ψ_1,…,Ψ_n))=H(Φ)+∑_k=1^nH(Ψ_k).
ER -