  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-Watson
estimator. When applied to a network, the best estimator near a vertex depends
on the amount of smoothness at the vertex. Often, there are no compelling
reasons to assume that a density will be continuous or discontinuous at a
vertex, hence a data driven approach is proposed. To estimate the density in a
neighborhood of a vertex, we propose a two-step 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 to model the change in slope. We study in detail the
special case of local piecewise linear regression and derive the leading bias
and variance terms using weighted least squares theory. We show that the
proposed approach will remove the bias near a vertex that has been noted for
existing methods, which typically do not allow for discontinuity at vertices.
For a fixed network, the proposed method scales sub-linearly with sample size
and it can be extended to regression and varying coefficient models on a
network. We demonstrate the workings of the proposed model by simulation
studies and apply it to a dendrite network data set.
