Nonparametric estimators over metric graphs

Abstract This work discusses a theory of functional spaces over metric graphs that permits the definition of penalized likelihood methods for data observed over spatial supports that are graphs. Within the considered mathematical framework, we recover classical results in functional analysis, such as a Poincaré-type inequality. This, in turn, enables us to uplift, to the present setting, the theory of some fundamental penalized likelihood methods. Specifically, we present two important classes of statistical models: nonparametric regression and nonparametric density estimation, here defined for data observed over graphs. We derive theoretical results regarding the well-posedness of the associated estimation problems and the consistency of the estimators. We also demonstrate the performance of the defined estimators with respect to state-of-the-art alternatives.