Density Estimation on a Network

This paper develops a novel approach to density estimation on a network. We formulate nonparametric density estimation on a network as a nonparametric regression problem by binning. Nonparametric regression using local polynomial kernel-weighted least squares have been studied rigorously, and its asymptotic properties make it superior to kernel estimators such as the Nadaraya-Waston estimator. Often, there are no compelling reasons to assume that a density will be continuous at a vertex and real examples suggest that densities often are discontinuous at vertices. To estimate the density in a neighborhood of a vertex, we propose a two-step local piecewise polynomial regression procedure. The first step of this pretest estimator fits a separate local polynomial regression on each edge using data only on that edge, and then tests for equality of the estimates at the vertex. If the null hypothesis is not rejected, then the second step re-estimates the regression function in a small neighborhood of the vertex, subject to a joint equality constraint. Since the derivative of the density may be discontinuous at the vertex, we propose a piecewise polynomial local regression estimate. We study in detail the special case of local piecewise linear regression and derive the leading bias and variance terms using weighted least squares matrix theory. We show that the proposed approach will remove the bias near a vertex that has been noted for existing methods.