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.

中文翻译：

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网络上的密度估计

本文开发了一种新的网络密度估计方法。我们通过分箱将网络上的非参数密度估计公式化为非参数回归问题。已经对使用局部多项式核加权最小二乘法的非参数回归进行了严格的研究，其渐近特性使其优于核估计器，例如 Nadaraya-Waston 估计器。通常，没有令人信服的理由假设密度在顶点处是连续的，真实的例子表明密度在顶点处通常是不连续的。为了估计顶点邻域的密度，我们提出了一个两步局部分段多项式回归程序。此预测试估计器的第一步仅使用该边上的数据在每条边上拟合单独的局部多项式回归，然后测试顶点估计的相等性。如果不拒绝原假设，则第二步在顶点的一个小邻域中重新估计回归函数，受联合等式约束。由于密度的导数在顶点处可能是不连续的，我们提出了分段多项式局部回归估计。我们详细研究了局部分段线性回归的特殊情况，并使用加权最小二乘矩阵理论推导出主要偏差和方差项。我们表明，所提出的方法将消除现有方法已注意到的顶点附近的偏差。由于密度的导数在顶点处可能是不连续的，我们提出了分段多项式局部回归估计。我们详细研究了局部分段线性回归的特殊情况，并使用加权最小二乘矩阵理论推导出主要偏差和方差项。我们表明，所提出的方法将消除现有方法已注意到的顶点附近的偏差。由于密度的导数在顶点处可能是不连续的，我们提出了分段多项式局部回归估计。我们详细研究了局部分段线性回归的特殊情况，并使用加权最小二乘矩阵理论推导出主要偏差和方差项。我们表明，所提出的方法将消除现有方法已注意到的顶点附近的偏差。