DEPENDENCE, DISPERSIVENESS, AND MULTIVARIATE HAZARD RATE ORDERING

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering. Also, it is shown that the marginal distributions of two upper orthant dispersive ordered random vectors are also dispersive ordered. Examples and applications are given.

[1]  Subhash C. Kochar,et al.  On tail-ordering and comparison of failure rates , 1986 .

[2]  A. Sampson,et al.  Uniform representations of bivariate distributions , 1975 .

[3]  K. Joag-dev,et al.  Bivariate Dependence Properties of Order Statistics , 1996 .

[4]  M. Shaked,et al.  Preservation of Stochastic Orderings under Random Mapping by Point Processes , 1995, Probability in the Engineering and Informational Sciences.

[5]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[6]  H. Wynn,et al.  Multivariate dispersion orderings , 1995 .

[7]  J. Bartoszewicz.  Moment inequalities for order statistics from ordered families of distributions , 1985 .

[8]  Albert W. Marshall,et al.  Some comments on the hazard gradient , 1975 .

[9]  Moshe Shaked,et al.  Two Variability Orders , 1998 .

[10]  Samuel Kotz,et al.  A vector multivariate hazard rate , 1975 .

[11]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[12]  J. Rojo,et al.  New properties and characterizations of the dispersive ordering , 1991 .

[13]  R. Nelsen.  An Introduction to Copulas , 1998 .

[14]  George Kimeldorf,et al.  One-parameter families of bivariate distributions with fixed marginals , 1975 .

[15]  Marco Scarsini,et al.  Stochastic Comparison of Random Vectors with a Common Copula , 2001, Math. Oper. Res..

[16]  GENERALIZED STOCHASTIC CONVEXITY AND STOCHASTIC ORDERINGS OF MIXTURES , 1999 .

[17]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[18]  Alfonso Suárez-Llorens,et al.  A multivariate dispersion ordering based on quantiles more widely separated , 2003 .

[19]  Moshe Shaked,et al.  Multivariate hazard rate orders , 2003 .

[20]  N. L. Johnson,et al.  Continuous Multivariate Distributions: Models and Applications , 2005 .

[21]  Jarosław Bartoszewicz.  Dispersive ordering and the total time on test transformation , 1986 .

[22]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[23]  P. A. P. Moran,et al.  ON THE QUANTILES OF THE GAMMA AND F DISTRIBUTIONS , 1978 .