Uniform Inference for Kernel Density Estimators with Dyadic Data

Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators which take the form of U-process-like dyadic empirical processes. We provide uniform point estimation and distributional results for the dyadic kernel density estimator, giving valid and feasible procedures for robust uniform inference. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized $t$-process. A consistent variance estimator is introduced in order to obtain analogous results for the Studentized $t$-process, enabling the construction of provably valid and feasible uniform confidence bands for the unknown density function. A crucial feature of U-process-like dyadic empirical processes is that they may be "degenerate" at some or possibly all points in the support of the data, a property making our uniform analysis somewhat delicate. Nonetheless we show formally that our proposed methods for uniform inference remain robust to the potential presence of such unknown degenerate points. For the purpose of implementation, we discuss uniform inference procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction methods. We illustrate the empirical finite-sample performance of our robust inference methods in a simulation study. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.