The Correlation Function of a Copula with a Singular Part
- 1Department of Mathematics, University of Almería, 04120 Almería, Spain
Abstract
We study bivariate copulas that have a singular component on the main diagonal and are generated by a single univariate function, the correlation function. The resulting class is that of lower semilinear copulas; in this parametrisation, known facts about it become statements about a single function, which is the conditional probability that the two coordinates coincide. Our main result is that the canonical correlation operator of each copula in the class is unitarily equivalent to multiplication by its correlation function, the unitary operator being the identity minus the Hardy averaging operator. Consequently, the spectrum of the canonical correlation operator is the essential range of the correlation function and the maximal correlation is its essential supremum; canonical functions exist if and only if the correlation function is constant on a set of positive measure, and the maximal correlation is attained if and only if the correlation function equals its essential supremum on such a set. We also treat copulas with singular components supported on a curve, a family that contains all Marshall–Olkin copulas and whose admissibility is governed by the elasticity of the curve, and derive Bayes factors for testing the equality of two binomial proportions, in closed form for the Fréchet family and as one-dimensional integrals in general.
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BibTeX
@article{ubedaflores2026correlation,
title = {{The Correlation Function of a Copula with a Singular Part}},
author = {Manuel Úbeda-Flores},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100759},
url = {https://www.mdpi.com/2075-1680/15/10/759},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The Correlation Function of a Copula with a Singular Part
AU - Manuel Úbeda-Flores
PY - 2026
DA - 2026/10/11
JO - Axioms
DO - 10.3390/axioms15100759
UR - https://www.mdpi.com/2075-1680/15/10/759
AB - We study bivariate copulas that have a singular component on the main diagonal and are generated by a single univariate function, the correlation function. The resulting class is that of lower semilinear copulas; in this parametrisation, known facts about it become statements about a single function, which is the conditional probability that the two coordinates coincide. Our main result is that the canonical correlation operator of each copula in the class is unitarily equivalent to multiplication by its correlation function, the unitary operator being the identity minus the Hardy averaging operator. Consequently, the spectrum of the canonical correlation operator is the essential range of the correlation function and the maximal correlation is its essential supremum; canonical functions exist if and only if the correlation function is constant on a set of positive measure, and the maximal correlation is attained if and only if the correlation function equals its essential supremum on such a set. We also treat copulas with singular components supported on a curve, a family that contains all Marshall–Olkin copulas and whose admissibility is governed by the elasticity of the curve, and derive Bayes factors for testing the equality of two binomial proportions, in closed form for the Fréchet family and as one-dimensional integrals in general.
SN - 2075-1680
ER -