Title: Minimax Optimal Rates of Estimation in Functional ANOVA Models with Derivatives,
Abstract: 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.