分层阿基米德连接函数的多面体框架：估计、层级恢复与奇异渐近

中文摘要 AI 辅助

分层阿基米德连接函数（HACs）允许冗余的树表示：当相邻内部节点的参数相等时，收缩连接边不会改变连接函数。这造成了结构上的过度参数化。对于齐次单参数族，我们消除了这种冗余，并将所得简化模型类与超度量扇形（ultrametric fan）等同起来。我们处理了Clayton、Frank、Ali-Mikhail-Haq、Gumbel和Joe族。使用归一化Kendall's tau的负logit变换作为共同的节点高度坐标，简化层级形成多面体层（polyhedral strata），二元层级索引极大锥（maximal cones），而多叉层级位于共同的收缩面上。诱导的共表型度量（cophenetic metric）生成与连接函数一致收敛相同的拓扑。我们还表明，信息丰富的成对依赖性摘要以分量方式作用于共表型坐标，从而为简化模型提供全局坐标。该表示产生一个全局估计器，通过反转估计的成对摘要并投影到扇形上获得，无需预先选择层级。我们证明了相合性和收敛速率的传递。在二元真实值下，未阈值化的估计器以趋于一的概率恢复层级；在多叉真实值下，它渐近地选择一个二元细化，而消失的阈值化一致地恢复简化层级。在成对摘要的联合中心极限定理下，投影估计器具有根号n极限，由高斯向量到切扇形（tangent fan）的欧几里得投影给出。该极限在二元层级处是高斯的，在多叉层级处通常是非高斯的。对于经验Kendall's tau，我们验证了所需的正则性条件并获得了相应的奇异极限。

英文摘要

Hierarchical Archimedean copulas (HACs) admit redundant tree representations: when adjacent internal nodes have equal parameters, contracting the connecting edge leaves the copula unchanged. This creates a structural overparametrization. For homogeneous one-parameter families, we remove this redundancy and identify the resulting reduced model class with the ultrametric fan. We treat the Clayton, Frank, Ali-Mikhail-Haq, Gumbel and Joe families. Using the negative logit of normalized Kendall's tau as a common node-height coordinate, reduced hierarchies form polyhedral strata, binary hierarchies index maximal cones, and multifurcating hierarchies lie on common contraction faces. The induced cophenetic metric generates the same topology as uniform convergence of copulas. We also show that informative pairwise dependence summaries act componentwise on cophenetic coordinates and hence provide global coordinates for the reduced model. This representation yields a global estimator obtained by inverting estimated pairwise summaries and projecting onto the fan, without preselecting a hierarchy. We prove consistency and convergence-rate transfer. At binary truths, the unthresholded estimator recovers the hierarchy with probability tending to one; at multifurcating truths, it asymptotically selects a binary refinement, while vanishing thresholding consistently recovers the reduced hierarchy. Under a joint central limit theorem for the pairwise summaries, the projected estimator has a root-$n$ limit given by Euclidean projection of a Gaussian vector onto the tangent fan. The limit is Gaussian at binary hierarchies and generally non-Gaussian at multifurcating ones. For empirical Kendall's tau, we verify the required regularity conditions and obtain the corresponding singular limit.

发表机构

• Alma Mater Studiorum University of Bologna（博洛尼亚大学）

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