Journal of the
Korean Mathematical Society JKMS

A random vector $\underline{X} =(X_1, \cdots, X_n )$ is conditionally weakly associated if and only if for every pair of partitions $ \underline{X_1} =(X_{\pi(1)}, \cdots, X_{\pi(k)} ),$ $ \underline{X_2} = (X_{\pi(k+1)}, \cdots, X_{\pi(n)})$ of $ \underline{X} ~P(\underline{X_1} \in A | \underline{X_2} \in B, \theta \in I) \ge P(\underline{X_1} \in A | \theta \in I )$ whenever A and B are open upper sets and $ \pi $ is any permutation of $ \{1, \cdots, n \} $. In this note we develop some concepts of conditional positive dependence, which are weaker than conditional weak association but stronger than conditional positive orthant dependence, by requiring the above inequality to hold only for some upper sets and applying the arguments in Shaked (1982).

Keywords: conditional positive dependence, conditional weak association, conditional positive orthant dependence, upper sets

MSC numbers: 62F10, 60E05