Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas
One of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, co...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/4/392 |
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| author | Christophe Chesneau |
| author_facet | Christophe Chesneau |
| author_sort | Christophe Chesneau |
| collection | DOAJ |
| description | One of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, contain a wide variety of dependent structures, and the copulas now in use may not provide the best model for all of them. As a result, researchers seek to innovate by building novel copulas with appealing properties that are also based on original methodologies. The foundations are theoretical; for a copula to be validated, it must meet specific requirements, which frequently dictate the constraints that must be placed on the relevant parameters. In this article, we make a contribution to the understudied field of one-variable-power copulas. We first identify the specific assumptions that, in theory, validate copulas of such nature. Some other general copulas and inequalities are discussed. Our general results are illustrated with numerous examples depending on two or three parameters. We also prove that strong connections exist between our assumptions and well-established distributions. To highlight the importance of our findings, we emphasize a particular two-parameter, one-variable-power copula that unifies the definition of some other copulas. We reveal its versatile shapes, related functions, various symmetry, Archimedean nature, geometric invariance, copula ordering, quadrant dependence, tail dependence, correlations, and distribution generation. Numerical tables and graphics are produced to support some of these properties. The estimation of the parameters based on data is discussed. As a complementary contribution, two new, intriguing one-variable-power copulas beyond the considered general form are finally presented and studied. |
| first_indexed | 2024-03-11T05:14:53Z |
| format | Article |
| id | doaj.art-52fe9ca80ec044a985c1fcd52f68e7e5 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T05:14:53Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-52fe9ca80ec044a985c1fcd52f68e7e52023-11-17T18:19:43ZengMDPI AGAxioms2075-16802023-04-0112439210.3390/axioms12040392Theoretical Validation of New Two-Dimensional One-Variable-Power CopulasChristophe Chesneau0Department of Mathematics, LMNO, Université de Caen-Normandie, 14032 Caen, FranceOne of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, contain a wide variety of dependent structures, and the copulas now in use may not provide the best model for all of them. As a result, researchers seek to innovate by building novel copulas with appealing properties that are also based on original methodologies. The foundations are theoretical; for a copula to be validated, it must meet specific requirements, which frequently dictate the constraints that must be placed on the relevant parameters. In this article, we make a contribution to the understudied field of one-variable-power copulas. We first identify the specific assumptions that, in theory, validate copulas of such nature. Some other general copulas and inequalities are discussed. Our general results are illustrated with numerous examples depending on two or three parameters. We also prove that strong connections exist between our assumptions and well-established distributions. To highlight the importance of our findings, we emphasize a particular two-parameter, one-variable-power copula that unifies the definition of some other copulas. We reveal its versatile shapes, related functions, various symmetry, Archimedean nature, geometric invariance, copula ordering, quadrant dependence, tail dependence, correlations, and distribution generation. Numerical tables and graphics are produced to support some of these properties. The estimation of the parameters based on data is discussed. As a complementary contribution, two new, intriguing one-variable-power copulas beyond the considered general form are finally presented and studied.https://www.mdpi.com/2075-1680/12/4/392one-variable-power copulasdependence modelsmultidimensional analysiscorrelation measures |
| spellingShingle | Christophe Chesneau Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas Axioms one-variable-power copulas dependence models multidimensional analysis correlation measures |
| title | Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas |
| title_full | Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas |
| title_fullStr | Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas |
| title_full_unstemmed | Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas |
| title_short | Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas |
| title_sort | theoretical validation of new two dimensional one variable power copulas |
| topic | one-variable-power copulas dependence models multidimensional analysis correlation measures |
| url | https://www.mdpi.com/2075-1680/12/4/392 |
| work_keys_str_mv | AT christophechesneau theoreticalvalidationofnewtwodimensionalonevariablepowercopulas |