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ORACLE-EFFICIENT NONPARAMETRIC ESTIMATION OF AN ADDITIVE MODEL WITH AN UNKNOWN LINK FUNCTION

Published online by Cambridge University Press:  11 October 2010

Joel L. Horowitz*
Affiliation:
Northwestern University
Enno Mammen
Affiliation:
University of Mannheim
*
*Address Correspondence to Joel L. Harowitz, Department of Economics, Northwestern University, Evanston, IL 60208-2600 U.S.A.; E-mail: joel-horowitz@northwestern.edu.

Abstract

This paper describes an estimator of the additive components of a nonparametric additive model with an unknown link function. When the additive components and link function are twice differentiable with sufficiently smooth second derivatives, the estimator is asymptotically normally distributed with a rate of convergence in probability of n−2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, the estimator has no curse of dimensionality. Moreover, the asymptotic distribution of the estimator of each additive component is the same as it would be if the link function and the other components were known with certainty. Thus, asymptotically there is no penalty for not knowing the link function or the other components.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2011

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ORACLE-EFFICIENT NONPARAMETRIC ESTIMATION OF AN ADDITIVE MODEL WITH AN UNKNOWN LINK FUNCTION
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