Minimax Optimal Rates of Estimation in Functional ANOVA Models with Derivatives

TL;DR

This paper derives the best possible convergence rates for estimating functions in functional ANOVA models when derivative data are available, showing derivatives can significantly improve estimation efficiency.

Contribution

It establishes minimax optimal rates for both function and derivative estimation in functional ANOVA models, highlighting the benefits of using derivative data.

Findings

Partial derivatives improve convergence rates for function estimation.

Optimal rates for full interaction models match those of lower interaction models when derivatives are used.

Using all first-order derivatives achieves parametric convergence rates in additive models.

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.