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A study on LTD and RTI positive dependence orderings

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  • Colangelo, Antonio

Abstract

In this paper we discuss the properties of the orderings of positive dependence introduced by Hollander et al. [Hollander, M., Proschan, F., Sconing, J., 1990. Information, censoring, and dependence. In: Topics in Statistical Dependence (Somerset, PA, 1987). In: IMS Lecture Notes Monogr. Ser., vol. 16. pp. 257-268. Inst. Math. Statist., Hayward, CA] as generalizations of the bivariate positive dependence concepts of left-tail decreasing (LTD) and right-tail increasing (RTI) random vectors studied by Esary and Proschan [Esary, J.D., Proschan, F., 1972. Relationships among some concepts of bivariate dependence. Ann. Math. Statist. 43, 651-655]. We show which of the postulates proposed by Kimeldorf and Sampson [Kimeldorf, G., Sampson, A.R., 1987. Positive dependence orderings. Ann. Inst. Statist. Math. 39 (1), 113-128] for a reasonable positive dependence ordering are satisfied and how the orders can be studied by restricting them to copulas. We also investigate the relationships of these orders with some other orderings which have appeared in the literature and generalize the same notions of positive dependence. Finally, some applications to extreme value bivariate distributions are discussed.

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  • Colangelo, Antonio, 2008. "A study on LTD and RTI positive dependence orderings," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2222-2229, October.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:14:p:2222-2229
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    1. Takemi Yanagimoto & Masashi Okamoto, 1969. "Partial orderings of permutations and monotonicity of a rank correlation statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 489-506, December.
    2. Z. Fang & H. Joe, 1992. "Further developments on some dependence orderings for continuous bivariate distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(3), pages 501-517, September.
    3. George Kimeldorf & Allan Sampson, 1989. "A framework for positive dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(1), pages 31-45, March.
    4. Colangelo, Antonio & Scarsini, Marco & Shaked, Moshe, 2005. "Some notions of multivariate positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 13-26, August.
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    1. Gijbels, Irène & Sznajder, Dominik, 2013. "Testing tail monotonicity by constrained copula estimation," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 338-351.

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