# Kernel Density Estimation for Undirected Dyadic Data

## TL;DR

This paper develops a kernel density estimation method for undirected dyadic data in networks, revealing that the estimator converges at the same rate as the sample mean despite local dependence.

## Contribution

It introduces a nonparametric kernel density estimator for dyadic network data and derives its asymptotic properties, including convergence rates and normality, under local dependence.

## Key findings

- Density estimates converge at the same rate as the sample mean (√N).
- The asymptotic distribution of the estimator is normal.
- The method accounts for local dependence in network data.

## Abstract

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.