A connection between supermodular ordering and positive/negative association
Date
2004ISSN
0047-259XSource
Journal of Multivariate AnalysisVolume
88Issue
1Pages
138-151Google Scholar check
Keyword(s):
Metadata
Show full item recordAbstract
In this paper, we show that a vector of positively/negatively associated random variables is larger/smaller than the vector of their independent duplicates with respect to the supermodular order. In that way, we solve an open problem posed by Hu (Chinese J. Appl. Probab. Statist. 16 (2000) 133) refering to whether negative association implies negative superadditive dependence, and at the same time to an open problem stated in Müller and Stoyan (Comparison Methods for Stochastic Modes and Risks, Wiley, Chichester, 2002) whether association implies positive supermodular dependence. Therefore, some well-known results concerning sums and maximum partial sums of positively/negatively associated random variables are obtained as an immediate consequence. The aforementioned result can be exploited to give useful probability inequalities. Consequently, as an application we provide an improvement of the Kolmogorov-type inequality of Matula (Statist. Probab. Lett. 15 (1992) 209) for negatively associated random variables. Moreover, a Rosenthal-type inequality for associated random variables is presented. © 2003 Elsevier Inc. All rights reserved.