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Positive Dependence and Weak Convergence

  • A. Colangelo (a1), A. Müller (a2) and M. Scarsini (a3)
Abstract

A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

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Copyright
Corresponding author
Postal address: Dipartimento di Economia, Università dell'Insubria, Via Monte Generoso 71, I-21100 Varese, Italy. Email address: antonio.colangelo@uninsubria.it
∗∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128 Karlsruhe, Germany. Email address: mueller@wior.uni-karlsruhe.de
∗∗∗ Postal address: Dipartimento di Statistica e Matematica Applicata, Università di Torino, Piazza Arbarello 8, I-10122 Torino, Italy. Email address: marco.scarsini@unito.it
References
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Alam, K. and Wallenius, K. T. (1976). Positive dependence and monotonicity in conditional distributions. Commun. Statist. Theory Meth. 5, 525534.
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765772.
Borgan, Ø., Hoem, J. M. and Norberg, R. (1981). A nonasymptotic criterion for the evaluation of automobile bonus systems. Scand. Actuarial J. 1981, 165178.
Colangelo, A., Scarsini, M. and Shaked, M. (2005). Some notions of multivariate positive dependence. Insurance Math. Econom. 37, 1326.
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899912.
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467498.
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.
Kimeldorf, G. and Sampson, A. R. (1987). Positive dependence orderings. Ann. Inst. Statist. Math. 39, 113128.
Kimeldorf, G. and Sampson, A. R. (1989). A framework for positive dependence. Ann. Inst. Statist. Math. 41, 3145.
Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 11371153.
Milgrom, P. R. and Weber, R. J. (1982). A theory of auctions and competitive bidding. Econometrica 50, 10891122.
Müller, A. (1997). Stochastic orders generated by integrals: a unified study. Adv. Appl. Prob. 29, 414428.
Müller, A. and Scarsini, M. (2001). Stochastic comparison of random vectors with a common copula. Math. Operat. Res. 26, 723740.
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.
Norberg, R. (1976). A credibility theory for automobile bonus systems. Scand. Actuarial J. 1976, 92107.
Pellerey, F. and Semeraro, P. (2003). A positive dependence notion based on the supermodular order. Res. Rep. 6, Dipartimento di Matematica, Politecnico di Torino.
Sarkar, S. K. (1998). Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Ann. Statist. 26, 494504.
Shiryaev, A. N. (1996). Probability, 2nd edn. Springer, New York.
Tukey, J. W. (1958). A problem of Berkson, and minimum variance orderly estimators. Ann. Math. Statist. 29, 588592.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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