Comparison results for positive supermodular dependent Markov tree distributions

Abstract

Positive dependencies have been compared in the literature under rather strong assumptions such as equality of conditional distributions, exchangeability, or stationarity. We establish supermodular ordering results for distributions that are Markov with respect to a tree structure. Our comparison results rely on simple stochastic monotonicity conditions and a pointwise ordering of bivariate copulas associated with the edges of the underlying tree. We also study flexibility of the marginal distributions in stochastic and convex order. As a consequence, we obtain first- and second-order stochastic dominance esults for extreme order statistics and sums of positively dependent random variables. As an application, we investigate distributional robustness of the maximum of a perturbed random walk under model uncertainty. Several examples and a detailed discussion of the assumptions demonstrate the generality of our results and reveal deeper insights into non-intuitive positive dependence properties of multidimensional distributions.