Open Access
December 2009 Estimation of functional derivatives
Peter Hall, Hans-Georg Müller, Fang Yao
Ann. Statist. 37(6A): 3307-3329 (December 2009). DOI: 10.1214/09-AOS686


Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the nature of the functional relationship, we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions. In statistical applications and data analysis, functional derivatives provide a quantitative measure of the often intricate relationship between changes in predictor trajectories and those in scalar responses. This approach provides a natural extension of classical gradient fields in vector space and provides directions of steepest descent. We suggest a kernel-based method for the nonparametric estimation of functional derivatives that utilizes the decomposition of the random predictor functions into their eigenfunctions. These eigenfunctions define a canonical set of directions into which the gradient field is expanded. The proposed method is shown to lead to asymptotically consistent estimates of functional derivatives and is illustrated in an application to growth curves.


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Peter Hall. Hans-Georg Müller. Fang Yao. "Estimation of functional derivatives." Ann. Statist. 37 (6A) 3307 - 3329, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2009

zbMATH: 1369.62064
MathSciNet: MR2549561
Digital Object Identifier: 10.1214/09-AOS686

Primary: 62G05 , 62G20

Keywords: consistency , Functional data analysis , gradient field , growth curve , Karhunen–Loève expansion , principal component , Small ball probability

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6A • December 2009
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