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Asymptotic results for conditional measures of association of a random sum
Institution:1. Cass Business School, City University, London EC1Y 8TZ, United Kingdom;2. College of Business and Public Administration, Drake University, 345 Aliber Hall, 2507 University Avenue, Des Moines, IA 50311, USA;1. Amsterdam School of Economics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, The Netherlands;2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada;1. Cass Business School, City University, London EC1Y 8TZ, United Kingdom;2. China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, 100081, China;1. University of Amsterdam, Netherlands;2. Riskcenter, University of Barcelona, Spain;1. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy;2. Dipartimento di Studi per l''Economia e l''Impresa, Università del Piemonte Orientale, Via Perrone 18, 28100 Novara, Italy;1. China Institute for Actuarial Science, Central University of Finance and Economics, China;2. KU Leuven, Leuven, Belgium;3. Faculty of Economics, Vrije Universiteit Brussel, Belgium
Abstract:Asymptotic results are obtained for several conditional measures of association. The chosen random variables are the first two order statistics and the total sum within a random sum. Many of the results have confirmed the “one-jump” property of the risk model. Non-trivial limits are obtained when the dependence among the first two order statistics is considered. Our results help in understanding the extreme behaviour of well-known reinsurance treaties that involve only few large claims. Interestingly, the Pearson product-moment correlation coefficient between the first two order statistics provides an alternative procedure to estimate the tail index of the underlying distribution.
Keywords:Extreme Value Theory  Long-tailed distribution  Gumbel tail  Kendall’s tau  Order statistics  Pearson product-moment correlation coefficient  Regular variation  Spearman’s rho
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