  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 taking the form of dyadic empirical
processes. 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 and Studentized
$t$-processes. A consistent variance estimator enables the construction of
valid and feasible uniform confidence bands for the unknown density function.
We showcase the broad applicability of our results by developing novel
counterfactual density estimation and inference methodology for dyadic data,
which can be used for causal inference and program evaluation. A crucial
feature of dyadic distributions is that they may be "degenerate" at certain
points in the support of the data, a property making our analysis somewhat
delicate. Nonetheless our methods for uniform inference remain robust to the
potential presence of such points. For implementation purposes, we discuss
inference procedures based on positive semi-definite covariance estimators,
mean squared error optimal bandwidth selectors and robust bias correction
techniques. We illustrate the empirical finite-sample performance of our
methods both in simulations and with real-world trade data, for which we make
comparisons between observed and counterfactual trade distributions in
different years. Our technical results concerning strong approximations and
maximal inequalities are of potential independent interest.
