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.

翻译：Let $1,\\ ldots,N$ 指数是一个简单的随机抽样单位。 对于每个单位,我们观察的是递归者的矢量 $X ⁇ i} 美元,对于每个单位,我们观察的是 $nleft(N-1\right) 订购的对数单位,结果$Y ⁇ ij} 美元。结果 $Y ⁇ ij} 美元 和 $Ykl} 美元是独立的, 如果它们的指数脱钩, 但却有其他依赖性( 即, “ 严重依赖性 ” ) 。让 最优化的( X ⁇ i}, X}}\\right) 美元 。 对于每个单位, 我们观察的递归回函数的矢量, 我们用抽样数据来计算出一个非参数性的估计值 $gleg\ left( N1\\\\\\\\\\\\rright) 。 当我们根据正统值计算到一个标准值时, 我们的递归缩率( ) 在一个标准下, 我们的正标值下, 显示一个正标值( ) 和正标值下, 显示一个正值的比值的比值, 显示一个正值 。