We are interested in measures of central tendency for a population on a network, which is modeled by a metric tree. The location parameters that we study are generalized Fr\'echet means obtained by minimizing the objective function $\alpha \mapsto E[\ell(d(\alpha,X))]$ where $\ell$ is a generic convex nondecreasing loss. We leverage the geometry of the tree and the geodesic convexity of the objective to develop a notion of directional derivative in the tree, which helps up locate and characterize the minimizers. Estimation is performed using a sample analog. We extend to a metric tree the notion of stickiness defined by Hotz et al. (2013), we show that this phenomenon has a non-asymptotic component and we obtain a sticky law of large numbers. For the particular case of the Fr\'echet median, we develop non-asymptotic concentration bounds and sticky central limit theorems.

翻译：本文研究网络群体中心趋势的度量方法，该网络以度量树为模型。我们通过最小化目标函数$\alpha \mapsto E[\ell(d(\alpha,X))]$（其中$\ell$为一般凸非递减损失函数）得到的广义Fréchet均值作为位置参数。利用树的几何结构和目标函数的测地凸性，我们提出了树中方向导数的概念，这有助于定位和刻画极小值点。通过样本模拟进行估计。我们将Hotz等人（2013）定义的黏滞性概念推广至度量树，证明该现象具有非渐近成分，并得到了黏滞性大数定律。针对Fréchet中位数的特例，我们推导了非渐近集中不等式和黏滞性中心极限定理。