# Minimax Risk and Uniform Convergence Rates for Nonparametric Dyadic Regression

Let i=1,…,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(N-1) 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=(X_i',X_j')'; using the sampled data we seek to construct a nonparametric estimate of the mean regression function g(W_ij)def≡𝔼[.Y_ij|X_i,X_j]. 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=dim(W_ij) influences the rate differently.

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