Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-07T14:51:30.667Z Has data issue: false hasContentIssue false
  • English
  • Français

Concepts of Setwise Dependence

Published online by Cambridge University Press:  27 July 2009

Devendra Chhetry ,
Allan R. Sampson  and
George Kimeldorf
Devendra Chhetry
Affiliation:
Department of Mathematics and StatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
Allan R. Sampson
Affiliation:
Department of Mathematics and StatisticsUniversity of Pittsburgh Pittsburgh, Pennsylvania 15260
George Kimeldorf
Affiliation:
Programs in Mathematical SciencesUniversity of Texas at Dallas Richardson, Texas 75080
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper motivates and introduces a number of new concepts of positive dependence among sets of random variables. In particular, setwise positive upper (and lower) orthant dependence, setwise association, and other related concepts are studied and their relationships are explored. These new concepts are applied to various multivariate normal distributions, some of which have both positive and negative covariances. A variety of new and interesting inequalities are also obtained.

Information

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 3 , July 1989 , pp. 367 - 380
Copyright
Copyright © Cambridge University Press 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ahmed, A.N., Léon, R., & Proschan, F. (1981). Generalized association with applications in rnultivariate statistics. Annals of statistics 96: 168–176.Google Scholar
Alam, K. & Saxena, K.M.L. (1981). Positive dependence in multivariate distributions. Communications in Statistics 10: 1183–1196.Google Scholar
Alam, K. & Wallenius, K.T. (1976). Positive dependence and monotonicity in conditional distributions. Communications in Statistics 5: 525–534.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing Probability models. Silver Springs, Maryland: To Begin With.Google Scholar
Block, H.W. & Ting, M.L. (1981). Some concepts of multivariate dependence. Communicaions in Statistics 10: 749–762.Google Scholar
Block, H.W., Savits, T.H., & Shaked, M. (1982). Some concepts of negative dependence. Annals of Probability 10: 765–772.CrossRefGoogle Scholar
Chhetry, D. & Kimeldorf, G. (1983). A note on positive dependence in multivariate distributions. Communications in Statistics 12 (22): 2601–2603.CrossRefGoogle Scholar
Chhetry, D., Kimeldorf, G., & Zahedi, H. (1986). Dependence structures in which uncorrelatedness implies independence. Statistics and Probability Letters 4: 197–201.CrossRefGoogle Scholar
Esary, J.D. & Proschan, F. (1972). Relationships among some concepts of bivariate dependence. Annals of Mathematical Statistics 43: 651–655.CrossRefGoogle Scholar
Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics 44: 1466–1474.CrossRefGoogle Scholar
Fuchs, C. & Sampson, A. (1987). Simultaneous confidence intervals for the general linear model. Biometrics 43: 457–469.CrossRefGoogle Scholar
Holland, P.W. & Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics 14: 1523–1543.CrossRefGoogle Scholar
Joag-Dev, K., Perlman, M.D., & Pitt, L.D. (1983). Association of normal random variables and Slepian's Inequality. Annals of Probability, 11: 451–455.CrossRefGoogle Scholar
Joag-Dev, K. & Proschan, F. (1983). Negative association of random variables, with applications. Annals of Mathematical Statistics 11: 286–293.Google Scholar
Jogdeo, K. (1982). Concepts of dependence. In Kotz, S. & Johnson, N., (eds.), Encyclopedia of statistical sciences, Vol. 2. New York: John Wiley & Sons, pp. 324–334.Google Scholar
Lehmann, E. (1966). Some concepts of dependence. Annals of Mathematical Statistics 73: 1137–1153.CrossRefGoogle Scholar
Marshall, A. & Olkin, I. (1979). Inequalities. Theory of majorization and its applications. New York: Academic Press.Google Scholar
Pitt, L.D. (1982). Positively correlated normal variables are associated. Annals of Probability 10: 496–499.CrossRefGoogle Scholar
Ruschendorf, L. (1981). Characterization of dependence concepts in normal distributions. Annals of the Institute of Statistical Mathematics Part A: 33: 347–359.CrossRefGoogle Scholar
Shaked, M. (1977). A concept of positive dependence for exchangeable random variables. Annals of Statistics 5: 505–515.CrossRefGoogle Scholar
Shaked, M. (1982). A general theory of some positive dependence notions. Journal of Multivariable Analysis 12: 199–218.CrossRefGoogle Scholar
Šidák, Z. (1971). On probabilities of rectangles in multivariate Student distributions: their dependence on correlations. Annals of Mathematical Statistics 42: 169–175.CrossRefGoogle Scholar
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Technical Journal 41: 463–501.CrossRefGoogle Scholar
Tong, Y.L. (1980). Probability inequalities in multivariate distributions. New York: Academic Press, Inc.Google Scholar