Deep Minimax Estimation for Nonparametric Instrumental Variables Regression

We study nonparametric instrumental-variable (IV) mean, expectile, and quantile regression using a common neural minimax estimator. ReLU networks represent the structural function and a bounded instrument-indexed critic. Estimation minimizes the largest absolute empirical residual-score moment without separately estimating a first-stage conditional law. Mean and expectile scores enter directly, whereas the quantile indicator is approximated by an integrated kernel. For exact empirical minimizers, we establish nonasymptotic bounds separating statistical error, approximation errors for both networks, and quantile-smoothing error. Under the stated regularity conditions and H\"older smoothness \(\zeta\), the expected adversarial-moment risk admits an upper bound of order \(n^-\zeta/(d_\max+2\zeta)\), up to logarithmic factors, where \(d_\max\) is the larger of the regressor and instrument dimensions. Additional localization, critic-link, and spectral conditions yield structural \(L^2(P_X)\) upper bounds. Simulation studies assess finite-sample structural recovery, while an application to randomized-encouragement diary data illustrates differences among fitted response curves.