无向并元数据的核密度估计
Kernel Density Estimation for Undirected Dyadic Data
摘要：
研究了无向并元随机变量的密度函数的非参数估计（即，为N阶加权网络中的所有n个\overset{def}{\equiv} \tbinom{N}{2}无序代理/节点对定义的随机变量）。这些随机变量满足局部相关性：网络中共享一个或两个索引的任何随机变量都可能是相关的，而那些没有共享共同索引的随机变量是独立的。在这种情况下，我们表明，密度函数可以估计Rosenblatt（1956）和Parzen（1962）的核估计方法的应用。我们建议的渐近方差估计的启发相结合的（i）Newey（1994）的核估计在“一元”设置的方差估计方法和（ii）的方差估计的（估计）密度的一个简单的网络首先提出的荷兰和Leinhardt（1976）。更不寻常的是我们的并矢密度估计的收敛速度和渐近（正态）分布。具体来说，我们表明，他们收敛在相同的速度（无条件）二进样本均值：节点的数量，N的平方根。这与一元数据的密度和回归函数的非参数估计的结果不同，一元数据的非参数估计的收敛速度通常比其相应的样本均值慢收敛速率。
英文摘要：
We study nonparametric estimation of density functions for undirected dyadic random variables (i.e., random variables defined for all n\overset{def}{\equiv}\tbinom{N}{2} unordered pairs of agents/nodes in a weighted network of order N). These random variables satisfy a local dependence property: any random variables in the network that share one or two indices may be dependent, while those sharing no indices in common are independent. In this setting, we show that density functions may be estimated by an application of the kernel estimation method of Rosenblatt (1956) and Parzen (1962). We suggest an estimate of their asymptotic variances inspired by a combination of (i) Newey's (1994) method of variance estimation for kernel estimators in the "monadic" setting and (ii) a variance estimator for the (estimated) density of a simple network first suggested by Holland and Leinhardt (1976). More unusual are the rates of convergence and asymptotic (normal) distributions of our dyadic density estimates. Specifically, we show that they converge at the same rate as the (unconditional) dyadic sample mean: the square root of the number, N, of nodes. This differs from the results for nonparametric estimation of densities and regression functions for monadic data, which generally have a slower rate of convergence than their corresponding sample mean.
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