Generate QR code
Open Access
Subscription Access
On a family of copulas partially coinciding with the Frechet–Hoeffding boundaries
https://doi.org/10.26907/0021-3446-2025-9-81-100
Abstract
The Grubbs's test statistics are studied, i.e. absolute values of extreme studentized deviations of n random observations from the mean. We consider the case when random observations have arbitrary continuous marginal distributions. The existence of two regions is proved; in one of them, the joint distribution function of these statistics is a linear function of their marginal distribution functions, and in the other, the joint distribution function is zero. We construct a Grubbs’s copula from the joint distribution of Grubbs’s statistics. For the case n>3, the existence of two domains within the unit square in which the Grubbs's copula coincides with the lower Frechet-Hoeffding boundary is proved. In the case of n=3, the Grubbs's copula is the Frechet-Hoeffding lower bound. The Grubbs's copula rotated by 180∘ also partially coincides with the Frechet-Hoeffding lower bound (in the case of n>3) and is the Frechet-Hoeffding lower bound (in the case of n=3). We prove that Grubbs's copulas rotated by 90∘ and 270∘ partially coincide with the Frechet-Hoeffding upper bound (in the case of n>3) and become the Frechet-Hoeffding upper bound (in the case n=3).
About the Author
L. K. Shiryaeva
Samara National Research University
Russian Federation
Russian Federation
Ludmila Konstantinovna Shiryaeva
34 Moskovskoe High., Samara, 443086
References
1. Genest C., Favre A.-C. Everything you always wanted to know about copula modelling but were afraid to ask, J. Hydrologic Eng. 12 (4), 347–368 (2007).
2. Bj¨ornham O., Br¨annstr¨om N., Persson L. Absolutely continuous copulas with prescribed support constructed by differential equations, with an application in toxicology, Commun. Stat. – Theory and Methods 51 (19), 6601–6625 (2022).
3. Jaworski P. On copulas with a trapezoid support, Depend. Model. 11 (1), 1–23 (2023).
4. Nelsen R.B. An introduction to copulas. 2nd ed. Springer Series in Statistics. (Springer, New York, 2006).
5. Fontanari A., Cirillo P., Oosterlee C.W. Lorenz-generated bivariate Archimedean copulas, Depend. Model. 8 (1), 186–209 (2020).
6. Durante F., Jaworski P. Absolutely continuous copulas with given diagonal sections, Commun. Stat. – Theory and Methods 37 (18), 2924–2942 (2008).
7. Jwaid T., De Baets B., Kalick´a J., Mesiar R. Conic aggregation functions, Fuzzy Sets Syst. 167, 3–20 (2011).
8. Dibala M., Vavr´ikov´a L. A note on Archimax copulas and their representation by means of conic copulas, Fuzzy Sets Syst. 308, 123–132 (2017).
9. Saminger-Platz S., Arias-Garc´ia J.J., Mesiar R., Klement E.P. Characterizations of bivariate conic, extreme value, and Archimax copulas, Depend. Model. 5 (1), 45–58 (2017).
10. Ширяева Л. К. О хвостовой зависимости для копула-функции Граббса, Изв. вузов. Матем. (12), 66–83 (2015).
11. Ширяева Л. К. О трехпараметрической копула-функции Граббса, Изв. вузов. Матем. (3), 54–71 (2019).
12. Busemann H., Feller W. Zur differentiation der Lebesgueschen integrale, Fundam. Math. 22, 226–256 (1934).
13. Durante F., Fern´andez-S´anchez J., Sempi C. A note on the notion of singular copula, Fuzzy Sets Syst. 211, 120–122 (2013).
14. Durante F., Foscolo E., Rodr´iguez-Lallena J., Ubeda-Flores M. ´ A method for constructing higher-dimensional copulas, Statistics 46 (3), 387–404 (2012).
15. Grubbs F. E. Procedures for detecting outlying observations in samples, Technometrics 11 (1), 1–21 (1969).
16. Zhang J., Yu K. The null distribution of the likelihood-ratio test for one or two outliers in a normal sample, TEST: An Official J. Spanish Soc. Stat. Oper. Res. 15 (1), 141–150 (2006).
17. Durante F., Sempi C. Copula theory: an introduction, Copula theory and its appl. 198, Lecture Notes in Statistics, 3–31 (Springer, Berlin, Heidelberg, 2010), URL : https://link.springer.com/book/10.1007/978-3-642-12465-5.
18. Brechmann V., Schepsmeier U. Modeling dependence with C- and D-Vine Copulas: The R package CDVine, J. Stat. Software 52 (3), 1–27 (2013).
19. Ширяева Л. К. О распределении статистик Граббса в случае нормальной выборки с выбросом, Изв. вузов. Матем. (4), 81–101 (2017).
20. de Amo E., Fern´andez-S´anchez J., Ubeda-Flores M. ´ Zero-sets of copulas, Fuzzy Sets Syst. 393, 143–159 (2020).
Review
For citations:
Shiryaeva L.K. On a family of copulas partially coinciding with the Frechet–Hoeffding boundaries. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;1(9):81-100. (In Russ.) https://doi.org/10.26907/0021-3446-2025-9-81-100
Views:
68
Email this article 