Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.
Mots clés : additive process, extremal process, limit theorems, pareto distribution
@article{PS_2014__18__667_0,
author = {Mitov, Kosto V. and Nadarajah, Saralees},
title = {Extremal and additive processes generated by {Pareto} distributed random vectors},
journal = {ESAIM: Probability and Statistics},
pages = {667--685},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2014001},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2014001/}
}
TY - JOUR AU - Mitov, Kosto V. AU - Nadarajah, Saralees TI - Extremal and additive processes generated by Pareto distributed random vectors JO - ESAIM: Probability and Statistics PY - 2014 SP - 667 EP - 685 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014001/ DO - 10.1051/ps/2014001 LA - en ID - PS_2014__18__667_0 ER -
%0 Journal Article %A Mitov, Kosto V. %A Nadarajah, Saralees %T Extremal and additive processes generated by Pareto distributed random vectors %J ESAIM: Probability and Statistics %D 2014 %P 667-685 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014001/ %R 10.1051/ps/2014001 %G en %F PS_2014__18__667_0
Mitov, Kosto V.; Nadarajah, Saralees. Extremal and additive processes generated by Pareto distributed random vectors. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 667-685. doi : 10.1051/ps/2014001. http://www.numdam.org/articles/10.1051/ps/2014001/
[1] and , Bayesian-inference for Pareto populations. J. Econom. 21 (1983) 287-306. | MR | Zbl
[2] and , Decomposition of multivariate extremal processes. Commun. Stat Theory Methods 25 (1996) 737-758. | MR | Zbl
[3] , Some outlier tests for multivariate samples. South African Stat. J. 13 (1979) 29-52. | Zbl
[4] , and , Regular Variation. Cambridge University Press, Cambridge (1987). | MR | Zbl
[5] , Multivariate extremes at work for portfolio risk measurement. Finance 23 (2002) 125-144.
[6] , and , Analysis of bivariate tail dependence using extreme value copulas: An application to the SOA medical large claims database. Belgian Actuarial Bull. 3 (2003) 33-41.
[7] and , Statistical methods for multivariate extremes: An application to structural design (with discussion). J. Appl. Stat. 43 (1994) 1-48. | Zbl
[8] and , The t copula and related copulas. Int. Stat. Rev. 73 (2005) 111-129. | Zbl
[9] and , Dynamic copula models for multivariate high-frequency data in finance. Working Paper, ETH Zurich (2003).
[10] and , Self-similar Processes. Princeton University Press, Princeton (2002). | Zbl
[11] and , Reliability evaluation for a multi-state system under stress-strength setup. Commun. Stat. Theory Methods 40 (2011) 547-558. | MR | Zbl
[12] and , Markov chain models for extreme wind speeds. Environmetrics 17 (2006) 795-809. | MR
[13] , Order statistics of samples from multivariate distributions. J. Amer. Stat. Assoc. 70 (1975) 674-680. | MR | Zbl
[14] and , Measure of financial risk using conditional extreme value copulas with EVT margins. J. Risk 11 (2009) 51-85.
[15] and , Table of Integrals, Series, and Products, 7th edition. Academic Press, San Diego (2007). | MR | Zbl
[16] , Fitting bivariate cumulative returns with copulas. Comput. Stat. Data Anal. 45 (2004) 355-372. | MR
[17] and , Maxima of normal random vectors: Between independence and complete dependence. Stat. Probab. Lett. 7 (1989) 283-286. | MR | Zbl
[18] , Latent structure models applied to the joint distribution of drivers' injuries in road accidents. Stat. Neerlandica 31 (1977) 105-111.
[19] and (1977). Mathematical-models for describing clustering of sociopathy and hysteria in families. British J. Psychiatry 130 294-297.
[20] , and , On the frequency of large stock market returns: Putting booms and busts into perspective. Rev. Econ. Stat. 23 (1991) 18-24.
[21] , Estimation of risk measures in energy portfolios using modern copula techniques. Discussion Paper No. 43, Dortmund (2012).
[22] , and , Tail risk of multivariate regular variation. Methodology Comput. Appl. Probab. 13 (2011) 671-693. | MR | Zbl
[23] , and , Bivariate threshold methods for extremes. J. R. Stat. Soc. B 54 (1992) 171-183. | MR | Zbl
[24] , Measuring extreme cross-market dependence for risk management: The case of Jamaican equity and foreign exchange markets. Financial Stability Department, Research and Economic Program. Division, Bank of Jamaica (2004).
[25] and , Estimating the probability of two dependent catastrophic events. ASTIN Colloquium. International Acturial Association, Brussels, Belgium (2004).
[26] , and , Extreme dependence of multivariate catastrophic losses. Scandinavian Actuarial J. (2006) 203-225. | MR | Zbl
[27] , Global financial risks, CVaR and contagion management. J. Business Policy Res. 7 (2012) 115-130.
[28] and , Multivariate distributions for the life lengths of components of a system sharing a common environment. J. Appl. Probab. 23 (1986) 418-431. | MR | Zbl
[29] and , The t copula with multiple parameters of degrees of freedom: Bivariate characteristics and application to risk management. Quant. Finance 10 (2010) 1039-1054. | MR | Zbl
[30] , , , and , Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrological Proc. 27 (2013) 1175-1190.
[31] , Multivariate Pareto distributions. Ann. Math. Stat. 33 (1962) 1008-1015. | MR | Zbl
[32] , and , A subjective Bayesian approach to the theory of queues I Modeling. Queueing Systems 1 (1987) 317-333. | MR | Zbl
[33] , and , Coupled continuous time random walks in finance. Phys. A: Stat. Mech. Appl. 370 (2006) 114-118. | MR
[34] and , Limit Distributions for Sums of Independent Random Vectors: Heavy Tails Theory Practice. Wiley, New York (2001). | MR | Zbl
[35] and , Portfolio modeling with heavy tailed random vectors, in Handbook of Heavy-Tailed Distributions in Finance, edited by S.T. Rachev. Elsevier, New York (2003) 595-640.
[36] and , Improving financial risk assessment through dependency. Stat. Model. 2 (2002) 103-122. | MR | Zbl
[37] , and , Approximation of aggregate and extremal losses within the very heavy tails framework. Technical Report, University of Karlsrhue, University of California, Santa Barbara, submitted to Quant. Finance (2008). | MR | Zbl
[38] , and , On the performance of a new bivariate pseudo Pareto distribution with application to drought data. Stochastic Environmental Research and Risk Assessment 26 (2011) 925-945.
[39] and , Probabilistic landslide hazard assessment using Copula modeling technique. Landslides 11 (2013) 565-573.
[40] and , Application of copula modelling to the performance assessment of reconstructed watersheds. Stochastic Environmental Research and Risk Assessment 26 (2013) 189-205.
[41] and , Regional air quality conformity in transportation networks with stochastic dependencies: A theoretical copula-based model. Networks and Spatial Economics 13 (2013) 373-397. | MR
[42] and , A functional extremal criterion. J. Math. Sci. 121 (2004) 2636-2644. | MR | Zbl
[43] and , Functional transfer theorems for maxima of iid random variables. C. R. Acad. Bulgare Sci. 57 (2004b) 9-14. | MR | Zbl
[44] , and , Random time-changed extremal processes. Theory Probab. Appl. 51 (2006) 752-772. | MR | Zbl
[45] , and , Sum and extremal processes over explosion area. C. R. Acad. Bulgare Sci. 59 (2006) 19-26. | MR | Zbl
[46] , and , Relationship between extremal and sum processes generated by the same point process. Serdika 35 (2009) 169-194. | MR | Zbl
[47] and , Multivariate Pareto distributions: Inference and financial applications. Commun. Stat. Theory Methods 39 (2010) 1013-1025. | MR | Zbl
[48] , Multivariate extreme value distributions (with a discussion). In: Proc. of the 43rd Session of the International Statistical Institute, Bull. Int. Stat. Institute 49 (1981) 859-878, 894-902. | MR | Zbl
[49] , and , Integrals and Series, vols. 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam (1986). | MR | Zbl
[50] and , Stable Paretian Models in Finance. Wiley, Chichester (2000). | Zbl
[51] and , Measures of tail asymmetry for bivariate copulas. Stat. Papers 54 (2013) 709-726. | MR
[52] , Bivariate extreme value theory: Models and estimation. Biometrika 75 (1988) 397-415. | MR | Zbl
[53] and , Temporal and spatial variability of drought in mountain catchments of the Nysa Klodzka basin. In: Climate Variability and Change - Hydrological Impacts, vol 308, Proc. of 15th FRIEND world conference held at Havana. Edited by S. Demuth, A. Gustard, E. Planos, F. Scatena and E. Servat. 308 (2006) 139-144.
[54] , and , A generalized beta copula with applications in modeling multivariate long-tailed data. Insurance: Math. Econ. 49 (2011) 265-284. | MR | Zbl
[55] , Dependence in target element detections induced by the environment. Naval Research Logistics 38 (1991) 567-577. | Zbl
[56] , The Gumbel mixed model applied to storm frequency analysis. Water Resources Management 14 (2000) 377-389.
[57] , , , and , Spatio-temporal variations of precipitation extremes in Xinjiang, China. J. Hydrology 434-435 (2012) 7-18.
Cité par Sources :
