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Density estimation on a network

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  • Liu, Yang
  • Ruppert, David

Abstract

A novel approach is proposed for density estimation on a network. Nonparametric density estimation on a network is formulated 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, a two-step procedure is proposed. 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, a piecewise polynomial local regression estimate is used to model the change in slope. The special case of local piecewise linear regression is studied in detail and the leading bias and variance terms are derived using weighted least squares theory. 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. The working of the proposed model is demonstrated by simulation studies and applications to a dendrite network dataset.

Suggested Citation

  • Liu, Yang & Ruppert, David, 2021. "Density estimation on a network," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    • Handle: RePEc:eee:csdana:v:156:y:2021:i:c:s016794732030219x
    DOI: 10.1016/j.csda.2020.107128
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    References listed on IDEAS

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    1. Adrian Baddeley & Aruna Jammalamadaka & Gopalan Nair, 2014. "Multitype point process analysis of spines on the dendrite network of a neuron," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 63(5), pages 673-694, November.
    2. Greg McSwiggan & Adrian Baddeley & Gopalan Nair, 2017. "Kernel Density Estimation on a Linear Network," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(2), pages 324-345, June.
    3. Hall, Peter & Wand, M. P., 1996. "On the Accuracy of Binned Kernel Density Estimators," Journal of Multivariate Analysis, Elsevier, vol. 56(2), pages 165-184, February.
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