Comparison results for positive supermodular dependent Markov tree distributions

TL;DR

The paper develops a comprehensive framework for ordering positive dependence in multivariate distributions that are Markovian along a tree. It proves a main supermodular ordering result: if edgewise copula specifications and stochastic monotonicity conditions along the tree satisfy certain SI/CI relations and bivariate comparisons (X_i,X_j) ≤sm (Y_i,Y_j), then the global vectors satisfy X ≤sm Y (with PSMD under additional assumptions). It further shows how marginal distributions can be flexibly ordered in stochastic or convex order while preserving these dependence orderings, yielding first/second-order dominance results for extrema and sums, and extends to distributional robustness results in hidden Markov models, including perturbed random walks. The work provides practical copula-based criteria that apply to Markov trees (including star and chain structures) and clarifies the limitations through counterexamples, highlighting the nuanced role of SI conditions. Overall, the results offer a versatile, model-agnostic approach to compare high-dimensional positive dependence and to derive robust performance bounds for functionals in applications like reliability, finance, and HMMs.

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.

Figures (8)

• Figure 1: Examples of Markov tree distributions: a) A Markov process in discrete time, where the underlying tree is a chain. b) Random variables $X_1,\ldots,X_d$ that are conditionally independent given the common factor variable $X_0,$ where the underlying tree has a star-like structure. c) A tree-indexed Markov process with a general underlying tree structure. d) A hidden Markov model with hidden nodes $(X_{i})_{i\in 2\mathbb N_0}$ and observable nodes $(X_i)_{i\in 2\mathbb N_0+1}$. Each model is uniquely determined by the univariate marginal distributions specifying the nodes and by the bivariate copulas $(B_{ij})_{(i,j)\in E}$ specifying the edges of the underlying tree $T=(N,E);$ see Proposition \ref{['prop: existence of markov tree dirstribution']}.
• Figure 2: An example that illustrates the positive dependence conditions \ref{['thm11']} and \ref{['thm12']} in Theorem \ref{['thm: supermodular order of tree specifications generalized']} for a tree $T=(N,E)$ on $12$ nodes with root $0$. An arrow $U\longrightarrow V$ indicates $V\uparrow_{st} U;$ similarly, an arrow $U\longleftrightarrow V$ indicates $V \uparrow_{st} U$ and $U\uparrow_{st} V.$ Note that there is no positive dependence condition between $X_0$ and $X_7$ and between $Y_3$ and $Y_4.$ The leaves consist of the set $L=\{2,4,5,6,8,10,11\}.$ The set $P\subseteq N$ consists of the leaf $\ell=4$ and the path $p(0,\ell)$ between the root $0$ and the leaf $\ell,$ i.e., $P=\{1,3,4\}.$ The root's child $k^*$ is given by $k^*=7$.
• Figure 3: The graphs illustrate the SI conditions on the hidden Markov processes $(X,X^*)=(X_n,X_n^*)_{n\in \mathbb{N}_0}$ and $(Y,Y^*)=(Y_n,Y_n^*)_{n\in \mathbb{N}_0}$ which lead to the comparison results in Corollary \ref{['thmHMM']}, where an arrow $U \to V$ indicates $V\uparrow_{st} U.$ The root of the underlying tree corresponds to the variable $X_0$ and $Y_0,$ respectively. Note that only the SI condition between $X_0$ and $X_0^*$ can be dropped, see Proposition \ref{['propSIass']}.
• Figure 4: Uncertainty bands of the distribution function $t\mapsto F_{\max\{X_0^*,\ldots,X_d^*\}}(t)$ in the classes of hidden Markov models considered in Theorem \ref{['thedisrw']} and Examples \ref{['exClaycem']} and \ref{['exsClayton']}, where $d = 200$ and $\sigma_n = 3$ (left plot) resp. $\sigma_n = 0.3 n$ (right plot). The upper (red) graph is the distribution function of $\max_{n\leq d}\{\underline{X}_n^*\}$, which corresponds to the hidden Markov model without observation error. The black/orange/blue graph corresponds to the distribution function of $\max_{n\leq d}\{\overline{X}^*_n\}$ in the Gaussian/Clayton/survival Clayton setting, considering maximal uncertainty in Example \ref{['exGaussobs']}/\ref{['exClaycem']}/\ref{['exsClayton']}.
• Figure 5: Samples of a Gaussian copula (left), Clayton copula (mid), and survival Clayton copula (right), each having Kendall's tau value $\tau = 0.795$ which corresponds to the parameter $\rho=\sqrt{9/10}$ for the Gaussian copula and $\theta=7.764$ for the Clayton and survival Clayton copula. The plots indicate that the Clayton copulas exhibit lower tail-dependencies, while the survival Clayton copulas have upper tail-dependencies.

Theorems & Definitions (41)

• Proposition 1.1: Supermodular ordering of Markov processes
• Lemma 1.2: Supermodular ordering of Markovian star structures
• Theorem 1.3: Supermodular ordering of Markov tree distributions
• Corollary 1.4: Supermodular ordering based on bivariate tree specifications
• Remark 1.5
• Definition 2.1: Directed tree
• Definition 2.2: Markov tree dependence; meeuwissen1994tree
• Definition 2.3: Bivariate tree specification; meeuwissen1994tree
• Proposition 2.4: Markov realization of a bivariate tree specification
• Definition 2.5: Stochastic orderings