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This paper considers adaptive, minimax estimation of a quadratic functionalin a nonparametric instrumental variables (NPIV) model, which is an importantproblem in optimal estimation of a nonlinear functional of an ill-posed inverseregression with an unknown operator. We first show that a leave-one-out, sieveNPIV estimator of the quadratic functional can attain a convergence rate thatcoincides with the lower bound previously derived in Chen and Christensen[2018]. The minimax rate is achieved by the optimal choice of the sievedimension (a key tuning parameter) that depends on the smoothness of the NPIVfunction and the degree of ill-posedness, both are unknown in practice. We nextpropose a Lepski-type data-driven choice of the key sieve dimension adaptive tothe unknown NPIV model features. The adaptive estimator of the quadraticfunctional is shown to attain the minimax optimal rate in the severelyill-posed case and in the regular mildly ill-posed case, but up to amultiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.

Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models