Convex generalized Fr\'echet means in a metric tree

Abstract

We are interested in measures of central tendency for a population $\mu$ on a network, which is modeled by a metric tree. The location parameters that we study are generalized Fr\'echet means defined as minimizers of the objective function $\alpha \mapsto \mathbb E[\ell(d(\alpha,X)) - \ell(d(o,X))]$, where $\ell$ is a convex, strictly increasing loss, $X\sim \mu$ and $o$ is an arbitrary origin. 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 us locate and characterize the minimizers. We then extend to a metric tree the concept of stickiness defined by Hotz et al. (2013). When $\mu$ is nondegenerate, any Fr\'echet $\ell$-mean of $\mu$ is either sticky, one-sided partly sticky or two-sided partly sticky, depending on the number of vanishing directional derivatives. Estimation is performed using a sample analog. We establish laws of large numbers and central limit theorems that reveal distinct asymptotic behaviors of empirical means across these three regimes: collapse onto the population mean in the sticky case, partial collapse accompanied by one-sided fluctuations in the one-sided case, and two-sided fluctuations without collapse in the two-sided case. We further develop two consistent estimators of the stickiness regime, based respectively on pairwise collisions among empirical means and on empirical difference quotients. For the particular case of the Fr\'echet median, we develop distribution-free non-asymptotic confidence regions for the entire set of minimizers.