Supermodular ordering of Poisson and binomial random vectors by tree-based correlations

• Autorzy: Kızıldemir B. , Privault N.

Identyfikatory

DOI

10.19195/0208-4147.38.2.7

• Języki publikacji: EN

Abstrakty

EN

We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.

Słowa kluczowe

EN

stochastic ordering; supermodular functions; Möbius transform; Möbius inversion; binary trees; Poisson random vectors; binomial random vectors

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 2
• Strony: 385--405
• Opis fizyczny: Bibliogr. 13 poz., rys.

Twórcy

autor

Kızıldemir B.

• Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore

autor

Privault N.

• Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore

Bibliografia

• [1] N. Bäuerle, A. Blatter, and A. Müller, Dependence properties and comparison results for Lévy processes, Math. Methods Oper. Res. 67 (1) (2008), pp. 161-186.
• [2] M. Denuit, E. Frostig, and B. Levikson, Supermodular comparison of time-to-ruin random vectors, Methodol. Comput. Appl. Probab. 9 (1) (2007), pp. 41-54.
• [3] E. Frostig, Ordering ruin probabilities for dependent claim streams, Insurance Math. Econom. 32 (1) (2003), pp. 93-114.
• [4] E. Frostig and F. Pellerey, Supermodular comparison of dependence models and multivariate processes, with applications, in: Recent Advances in Probability and Statistics, Lect. Notes Semin. Interdiscip. Mat., Vol. 12, Potenza, 2015, pp 125-138.
• [5] T. Hu, C. Xie, and L. Ruan, Dependence structures of multivariate Bernoulli random vectors, J. Multivariate Anal. 94 (1) (2005), pp. 172-195.
• [6] K. Kawamura, The structure of multivariate Poisson distribution, Kodai Math. J. 2 (3) (1979), pp. 337-345.
• [7] B. Kızıldemir and N. Privault, Supermodular ordering of Poisson arrays, Statist. Probab. Lett. 98 (2015), pp. 136-143.
• [8] M. A. Meyer and B. Strulovici, The supermodular stochastic ordering, CEPR Discussion Paper DP9486, May 2013. http://www.cepr.org/pubs/dps/DP9486.
• [9] A. Müller, Stop-loss order for portfolios of dependent risks, Insurance Math. Econom. 21 (3) (1997), pp. 219-223.
• [10] A. Müller and M. Scarsini, Some remarks on the supermodular order, J. Multivariate Anal. 73 (1) (2000), pp. 107-119.
• [11] A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, Chichester 2002.
• [12] G. Peccati and M. S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementation. Supplementary Material Available Online, Springer, Milan 2011.
• [13] G. Rota, On the foundations of combinatorial theory: I. Theory of Möbius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), pp. 340-368.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).