Optimal Estimation in Functional ANOVA Models with Derivatives

• Activity Number: 333 - Section on Nonparametric Statistics - Student Paper Awards
• Type: Topic Contributed
• Date/Time: Tuesday, July 31, 2018 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Xiaowu Dai* and Peter Chien
• Companies: University of Wisconsin Madison and University of Wisconsin-Madison
• Keywords: Nonparametric regression; smoothing spline ANOVA; partial derivative data; method of regularization; minimax rate

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 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.