Minimax Risk and Uniform Convergence Rates for Nonparametric Dyadic Regression

TL;DR

This paper derives minimax risk bounds and establishes convergence rates for nonparametric dyadic regression, showing that the Nadaraya-Watson estimator achieves optimal rates under dyadic dependence, which differ from iid cases.

Contribution

It provides the first minimax risk bounds and convergence rates for nonparametric dyadic regression, highlighting the impact of dyadic dependence on estimation.

Findings

Nadaraya-Watson estimator achieves optimal convergence rates with appropriate bandwidths.

Effective sample size is smaller under dyadic dependence, affecting convergence rates.

Optimal rates differ from iid data due to dependence structure and dimension of covariates.

Abstract

Let $i=1,\ldots,N$ index a simple random sample of units drawn from some large population. For each unit we observe the vector of regressors $X_{i}$ and, for each of the $N\left(N-1\right)$ ordered pairs of units, an outcome $Y_{ij}$. The outcomes $Y_{ij}$ and $Y_{kl}$ are independent if their indices are disjoint, but dependent otherwise (i.e., "dyadically dependent"). Let $W_{ij}=\left(X_{i}',X_{j}'\right)'$; using the sampled data we seek to construct a nonparametric estimate of the mean regression function $g\left(W_{ij}\right)\overset{def}{\equiv}\mathbb{E}\left[\left.Y_{ij}\right|X_{i},X_{j}\right].$ We present two sets of results. First, we calculate lower bounds on the minimax risk for estimating the regression function at (i) a point and (ii) under the infinity norm. Second, we calculate (i) pointwise and (ii) uniform convergence rates for the dyadic analog of the familiar Nadaraya-Watson (NW) kernel regression estimator. We show that the NW kernel regression estimator achieves the optimal rates suggested by our risk bounds when an appropriate bandwidth sequence is chosen. This optimal rate differs from the one available under iid data: the effective sample size is smaller and $d_W=\mathrm{dim}(W_{ij})$ influences the rate differently.