  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.
