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BY 4.0 license Open Access Published by De Gruyter Open Access July 4, 2020

The deFinetti representation of generalised Marshall–Olkin sequences

  • Henrik Sloot EMAIL logo
From the journal Dependence Modeling

Abstract

We show that each infinite exchangeable sequence τ1, τ2, . . . of random variables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

MSC 2010: 60G09; 60G51; 62H05

References

[1] Aldous, D. J. (1985). Exchangeability and Related Topics. Springer, Berlin.10.1007/BFb0099421Search in Google Scholar

[2] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus. Second edition. Cambridge University Press.10.1017/CBO9780511809781Search in Google Scholar

[3] Berg, C., J. P. R. Christensen, and P. Ressel (1984). Harmonic Analysis on Semigroups, Springer, New York.10.1007/978-1-4612-1128-0Search in Google Scholar

[4] Bernhart, G., J.-F. Mai, and M. Scherer (2015). On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions. Depend. Model. 3, 29–46.10.1515/demo-2015-0003Search in Google Scholar

[5] Bertoin, J. (1999). Subordinators: Examples and Applications. Springer, Berlin.10.1007/978-3-540-48115-7_1Search in Google Scholar

[6] de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7, 1–68.Search in Google Scholar

[7] Embrechts, P. and M. Hofert (2013). A note on generalized inverses. Math. Methods Oper. Res. 77(3), 423–432.10.1007/s00186-013-0436-7Search in Google Scholar

[8] Gnedin, A. and J. Pitman (2008). Moments of convex distribution functions and completely alternating sequences. In D. Nolan and T. Speed (Eds.), Probability and statistics: essays in honor of David A. Freedman., pp. 30–41, Inst. Math. Statist., Beachwood OH.Search in Google Scholar

[9] Li, X. and F. Pellerey (2011). Generalized Marshall–Olkin distributions and related bivariate aging properties. J. Multivariate Anal. 102(10), 1399–1409.10.1016/j.jmva.2011.05.006Search in Google Scholar

[10] Lin, J. and X. Li (2014). Multivariate generalized Marshall–Olkin distributions and copulas. Methodol. Comput. Appl. Probab. 16(1), 53–78.10.1007/s11009-012-9297-4Search in Google Scholar

[11] Mai, J.-F. (2010). Extendibility of Marshall–Olkin Distributions via Lévy Subordinators and an Application to Portfolio Credit Risk. PhD thesis, Technische Universität München, Germany.Search in Google Scholar

[12] Mai, J.-F. (2014). Multivariate Exponential Distributions with Latent Factor Structure and Related Topics. Habilitation thesis, Technische Universität München, Germany.Search in Google Scholar

[13] Mai, J.-F. (2019). The infinite extendibility problem for exchangeable real-valued random vectors. Available at: https://arxiv.org/abs/1907.04054.Search in Google Scholar

[14] Mai, J.-F., S. Schenk, and M. Scherer (2016). Exchangeable exogenous shock models. Bernoulli 22(2), 1278–1299.10.3150/14-BEJ693Search in Google Scholar

[15] Mai, J.-F., S. Schenk, and M. Scherer (2017). Two novel characterizations of self-decomposability on the half-line. J. Theoret. Probab. 30(1), 365–383.10.1007/s10959-015-0644-6Search in Google Scholar

[16] Mai, J.-F. and M. Scherer (2009a). E˛ciently sampling exchangeable Cuadras–Augé copulas in high dimensions. Inform. Sci. 179(17), 2872–2877.10.1016/j.ins.2008.09.004Search in Google Scholar

[17] Mai, J.-F. and M. Scherer (2009b). Lévy-frailty copulas. J. Multivariate Anal. 100(7), 1567–1585.10.1016/j.jmva.2009.01.010Search in Google Scholar

[18] Mai, J.-F. and M. Scherer (2013). Sampling exchangeable and hierarchical Marshall-Olkin distributions. Comm. Statist. Theory Methods 42(4), 619–632.10.1080/03610926.2011.615437Search in Google Scholar

[19] Mai, J.-F. and M. Scherer (2014). Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time. Extremes 17(1), 77–95.10.1007/s10687-013-0175-4Search in Google Scholar

[20] Mai, J.-F. and M. Scherer (2017). Simulating Copulas: Stochastic Models, Sampling Algorithms and Applications. Second edition. World Scientific.10.1142/10265Search in Google Scholar

[21] Marshall, A. W. and I. Olkin (1967). A multivariate exponential distribution. J. Amer. Statist. Assoc. 62(317), 30–44.10.1080/01621459.1967.10482885Search in Google Scholar

[22] Muliere, P. and M. Scarsini (1987). Characterization of a Marshall–Olkin type class of distributions. Ann. Inst. Statist. Math. 39(2), 429–441.10.1007/BF02491480Search in Google Scholar

[23] Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139(11), 3921–3927.10.1016/j.jspi.2009.05.030Search in Google Scholar

[24] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Search in Google Scholar

[25] Schenk, S. (2016). Exchangeable Exogenous Shock Models. PhD thesis, Technische Universität München, Germany.Search in Google Scholar

[26] Schilling, R. L., R. Song, and Z. Vondracek (2012). Bernstein Functions. Theory and Applications. Second edition. De Gruyter, Berlin.10.1515/9783110269338Search in Google Scholar

Received: 2020-01-28
Accepted: 2020-05-22
Published Online: 2020-07-04

© 2020 Henrik Sloot, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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