论文标题
网络时刻的Edgeworth扩展
Edgeworth expansions for network moments
论文作者
论文摘要
MOMENTS ARXIV的网络方法：1202.5101是非参数网络推断的重要工具。但是，几乎没有研究网络矩统计的采样分布的准确描述。在本文中，我们通过Edgeworth扩展向学生化网络时刻的采样CDF提出了第一个高阶精度近似。与无噪声U统计数据的古典文献形成鲜明对比的是，我们表明，Edgeworth作为嘈杂的U统计量的Edgeworth扩展可以在没有非务实或平滑度假设的情况下实现高阶精度，而只是需要较弱的规律性条件。这一结果的背后是我们令人惊讶的发现，网络分析中的两个典型因素，即稀疏性和边缘性观察错误，共同发挥了祝福作用，在网络矩统计中产生了至关重要的自动化效应，并使之在分析上可以进行分析。我们的假设符合相关文献中的最低要求。对于稀疏的网络，我们的理论显示了一个简单的正常近似值，随着网络变得更加稀疏，逐渐贬值的浆果 - 贝里（Berry-Esseen）结合。该结果还完善了以前最好的理论结果。 对于从业者来说，我们的经验Edgeworth扩展非常准确，快速且易于实施。我们通过全面的仿真研究证明了我们方法的明显优势。 我们在网络推理中展示了结果的三个应用程序。据我们所知，我们证明了某些网络引导程序方案的高阶准确性的第一个理论保证，此外，这是为网络子放样选择子样本大小选择子样本大小的第一个理论指南。我们还在给定的时刻中得出一个样本测试和康沃尔 - 法派置信区间，并分别具有更高阶段的置信水平和I型误差的更高阶段。
Network method of moments arXiv:1202.5101 is an important tool for nonparametric network inference. However, there has been little investigation on accurate descriptions of the sampling distributions of network moment statistics. In this paper, we present the first higher-order accurate approximation to the sampling CDF of a studentized network moment by Edgeworth expansion. In sharp contrast to classical literature on noiseless U-statistics, we show that the Edgeworth expansion of a network moment statistic as a noisy U-statistic can achieve higher-order accuracy without non-lattice or smoothness assumptions but just requiring weak regularity conditions. Behind this result is our surprising discovery that the two typically-hated factors in network analysis, namely, sparsity and edge-wise observational errors, jointly play a blessing role, contributing a crucial self-smoothing effect in the network moment statistic and making it analytically tractable. Our assumptions match the minimum requirements in related literature. For sparse networks, our theory shows a simple normal approximation achieves a gradually depreciating Berry-Esseen bound as the network becomes sparser. This result also refines the best previous theoretical result. For practitioners, our empirical Edgeworth expansion is highly accurate, fast and easy to implement. We demonstrate the clear advantage of our method by comprehensive simulation studies. We showcase three applications of our results in network inference. We prove, to our knowledge, the first theoretical guarantee of higher-order accuracy for some network bootstrap schemes, and moreover, the first theoretical guidance for selecting the sub-sample size for network sub-sampling. We also derive one-sample test and Cornish-Fisher confidence interval for a given moment with higher-order accurate controls of confidence level and type I error, respectively.
