Abstract
We consider one-sided Grubbs's statistics for a normal sample of the size n. These statistics are extreme studentized deviations of the observations from the sample mean. One abnormal observation (outlier) is assumed in the sample, its number is unknown. We consider the case when the outlier differs from other observations in values of population mean and dispersion, i. e., shift and scale parameters. We construct a copula-function by an inversion method from the joint distribution of Grubbs's statistics, it depends on three parameters: shift and scale parameters and n. It is proved that for Grubbs's copula-function, the coefficients of the upper-left and lower-right tail dependencies are equal each other. Moreover, their value is independent of the shift and scale parameters but it depends on parameter n. The dependence in the tails of the distribution of the three-parameter Grubbs's copula coincides with the dependence in the tails of the joint distribution of one-sided Grubbs's statistics calculated from the normal sample without outlier.
Similar content being viewed by others
REFERENCES
Fantazzini, D. “Modeling of Multi-dimensional Distributions using Copula-functions. I”, Prikladnaya Ekonometrika 22 (2), 98–134 (2011).
Shiryaeva, L.K. “On the Distribution of Grubbs Statistics in the Case of a Normal Sample with an Outlier”, Russian Math. (Iz. VUZ) 61 (4), 72–88 (2017).
Pearson, E.S., Chandra Secar C. “The Efficiency of Statistical Tools and a Criterion for the Rejection of Outlying Observations”, Biometrika 28, 308–320 (1936).
Grubbs, F. “Sample Criteria for Testing Outlying Observations”, Ann. Math. Statist. 21 (1), 27–58 (1950).
Barnett, V., Lewis, T. Outliers in Statistical Data (John Wiley & Sons, Chichester, 1984).
Zhang, J., Keming, Y. “The Null Distribution of the Likelihood-ratio Test for One or Two Outliers in a Normal Sample”, TEST 15 (1), 141–150 (2006).
Shiryaeva, L.K. “On the Null and Alternative Distributions of the Test Statistic for the Largest (with Respect to Absolute Value) Normed Deviation”, Izv. Vyssh. Uchebn. Zaved. Mat. 10, 62–78 (2014).
Shiryaeva, L.K. “Using Special Hermite Functions to Study the Power Properties of the Grubbs Criterion”, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Ser. Fiz.-Matem. Nauki 29 (4), 131–145 (2012).
Nelsen, R.B. An Introduction to Copulas. Lecture Notes in Statistics (Springer-Verlag, New York, 2006).
Shiryaeva, L.K. “On a Three-parameter Grubbs Copula Function”, Russian Math. (Iz. VUZ) 63 (3), 45–61 (2019).
Shiryaeva, L.K. “On Tail Dependence for Grubbs' Copula-function”, Russian Math. (Iz. VUZ) 59 (12), 56–72 (2015).
Shiryaeva, L.K., Repina, E.G. “On Some Properties of Symmetric Grubbs Copula”, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Ser. Fiz.-Matem. Nauki 22 (4), 714–734 (2018).
Ammann, M., Suss, S. “Asymmetric Dependence Patterns in Financial Time Series”, The European J. Finance 15 (7–8), 703–719 (2009).
Joe, H. Multivariate Models and Dependence Concepts (Chapman Hall, London, 1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 12, pp. 51–73.
About this article
Cite this article
Shiryaeva, L.K. On Tail Dependence for Three-parameter Grubbs' Copula. Russ Math. 64, 46–66 (2020). https://doi.org/10.3103/S1066369X20120063
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X20120063