Convex generalized Fréchet means in a metric tree

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\échet 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\échet median, we develop non-asymptotic concentration bounds and sticky central limit theorems.