A Markov tree is a probabilistic graphical model for a random vector indexed by the nodes of an undirected tree encoding conditional independence relations between variables. One possible limit distribution of partial maxima of samples from such a Markov tree is a max-stable Hüsler–Reiss distribution whose parameter matrix inherits its structure from the tree, each edge contributing one free dependence parameter. Our central assumption is that, upon marginal standardization, the data-generating distribution is in the max-domain of attraction of the said Hüsler–Reiss distribution, an assumption much weaker than the one that data are generated according to a graphical model. Even if some of the variables are unobservable (latent), we show that the underlying model parameters are still identifiable if and only if every node corresponding to a latent variable has degree at least three. Three estimation procedures, based on the method of moments, maximum composite likelihood, and pairwise extremal coefficients, are proposed for usage on multivariate peaks over thresholds data when some variables are latent. A typical application is a river network in the form of a tree where, on some locations, no data are available. We illustrate the model and the identifiability criterion on a data set of high water levels on the Seine, France, with two latent variables. The structured Hüsler–Reiss distribution is found to fit the observed extremal dependence patterns well. The parameters being identifiable we are able to quantify tail dependence between locations for which there are no data.

马尔可夫树是用于随机向量的概率图形模型，该随机向量由对变量之间的条件独立性关系进行编码的无向树的节点索引。此类马尔可夫树中样本的部分最大值的一种可能的极限分布是最大稳定的Hüsler-Reiss分布，其参数矩阵从树继承其结构，每个边贡献一个自由相关性参数。我们的中心假设是，在进行边际标准化后，数据生成的分布处于所述Hüsler-Reiss分布的最大吸引域内，这一假设比根据图形模型生成数据的假设要弱得多。即使某些变量是不可观察的（潜在的），我们表明，当且仅当与潜在变量相对应的每个节点的度数至少为3时，基础模型参数仍然是可识别的。提出了基于矩量法，最大复合似然法和成对极值系数法的三种估计程序，用于在某些变量潜在时超过阈值数据的多元峰。典型的应用是树状的河网，其中在某些位置没有可用数据。我们用两个潜在变量说明了法国塞纳河高水位数据集的模型和可识别性标准。发现结构化的Hüsler-Reiss分布非常符合观察到的极端依赖模式。