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Biometrika Advance Access originally published online on February 4, 2008
Biometrika 2008 95(2):451-467; doi:10.1093/biomet/asm095
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© 2008 Biometrika Trust

Articles

Model diagnosis for parametric regression in high-dimensional spaces

W. Stute

Mathematical Institute, University of Giessen, D-35392 Giessen, Germany Winfried.Stute{at}math.uni-giessen.de

W. L. Xu

School of Statistics, Renmin University of China, Beijing, China wlxu{at}ruc.edu.cn

L. X. Zhu

Department of Mathematics, Hong Kong Baptist University, Hong Kong, China lzhu{at}hkbu.edu.hk

Received for publication 1 May 2006. Revision received 1 August 2007.
  Abstract

We study tools for checking the validity of a parametric regression model. When the dimension of the regressors is large, many of the existing tests face the curse of dimensionality or require some ordering of the data. Our tests are based on the residual empirical process marked by proper functions of the regressors. They are able to detect local alternatives converging to the null at parametric rates. Parametric and nonparametric alternatives are considered. In the latter case, through a proper principal component decomposition, we are able to derive smooth directional tests which are asymptotically distribution-free under the null model. The new tests take into account precisely the ‘geometry of the model’. A simulation study is carried through and an application to a real dataset is illustrated.

Key Words: Marked residual empirical process • Model check • Principal components


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