Minimax Optimal Rates of Estimation in Functional ANOVA Models with Derivatives

We establish minimax optimal rates of convergence for nonparametric estimation in functional ANOVA models when data from first-order partial derivatives are available. Our results reveal that partial derivatives can improve convergence rates for function estimation with deterministic or random designs. In particular, for full $d$-interaction models, the optimal rates with first-order partial derivatives on $p$ covariates are identical to those for $(d-p)$-interaction models without partial derivatives. For additive models, the rates by using all first-order partial derivatives are root-$n$ to achieve the "parametric rate". We also investigate the minimax optimal rates for first-order partial derivative estimations when derivative data are available. Those rates coincide with the optimal rate for estimating the first-order derivative of a univariate function.