Distribution of the Joint Survival Function of an Archimedean Copula

• Published in: Bulletin - Calcutta Statistical Association Vol. 76; no. 2; pp. 193 - 211
• Main Authors: Djimdou, Magloire Loudegui, Chaubey, Yogendra P., Sen, Arusharka
• Format: Journal Article
• Language: English
• Published: New Delhi, India SAGE Publications 01.11.2024
• Subjects: Archimedean copula; Kendall distribution; Kendall process
• ISSN: 0008-0683; 2456-6462; 2456-6462
• DOI: 10.1177/00080683241246438

Suppose a random vector U 1 , … , U d with values in the unit cube has a joint survival function: C * u 1 , … , u d = ℙ U 1 > u 1 , … , U d > u d , given by an Archimedean copula C u 1 , … , u d = φ − 1 φ u 1 + … + φ u d , with generator φ : 0 , 1 → 0 , ∞ , a smooth decreasing convex function such that φ 1 = 0 . In this article, we provide a formula for the distribution of Z = C * U 1 , … , U d = ℙ U 1 ' > U 1 , … , U d ' > U d ∣ U 1 , … , U d , where U 1 ' , … , U d ' is an independent copy of U 1 , … , U d and a method to simulate values from the distribution of Z in the bivariate case, that is, when d = 2. The case d > 2 does not seem to be tractable. As an application, we show how our result can be used to compute the limiting covariance of the empirical Kendall process corresponding to C * U 1 , U 2 . AMS Subject Classification: 62H05