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 Fréchet means obtained by minimizing the objective function
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 Fréchet median we develop non-asymptotic concentration bounds and sticky central limit theorems.