Biometrika Advance Access published online on February 4, 2008
Biometrika, doi:10.1093/biomet/asm095
Articles |
Model diagnosis for parametric regression in high-dimensional spaces
Mathematical Institute, University of Giessen, D-35392 Giessen, Germany Winfried.Stute{at}math.uni-giessen.de
School of Statistics, Renmin University of China, Beijing, China wlxu{at}ruc.edu.cn
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.
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
References
-
Behnen K., Neuhaus G. A central limit theorem under contiguous alternatives. Ann. Statist. (1975) 3:1349–53.[CrossRef]
Durbin J. Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. (1973) 1:279–90.[CrossRef]
Fan J., Huang L. Goodness-of-fit tests for parametric regression models. J. Am. Statist. Assoc. (2001) 96:640–52.[CrossRef][ISI]
Hart J. D. Nonparametric Smoothing and Lack-of-Fit Tests (1997) New York: Springer-Verlag.
He X., Ng P. Quantile splines with several covariates. J. Statist. Plan. Infer. (1999) 75:343–52.[CrossRef]
He X., Zhu L. X. A lack-of-fit test for quantile regression. J. Am. Statist. Assoc. (2003) 98:1013–22.[CrossRef][ISI]
Jennrich R. I. Asymptotic properties of non-linear least-squares estimators. Ann. Math. Statist. (1969) 40:633–43.[CrossRef]
Neyman J. Smooth test for goodness of fit. Skand. Aktuar. (1937) 20:149–99.
Pollard D. Convergence of Stochastic Processes (1984) New York: Springer-Verlag.
Shorack G. R., Wellner J. A. Empirical Processes with Applications to Statistics (1986) New York: Wiley.
Stute W. Nonparametric model checks for regression. Ann. Statist. (1997) 25:613–41.[CrossRef]
Stute W., Thies S., Zhu L. X. Model checks for regression: an innovation process approach. Ann. Statist. (1998) 26:1916–34.[CrossRef]
Stute W., Zhu L. X. Nonparametric checks for single-index models. Ann. Statist. (2005) 33:1048–83.[CrossRef]
White H. Consequences and detection of misspecified nonlinear regression models. J. Am. Statist. Assoc. (1981) 76:419–33.[CrossRef][ISI]
Zhu L. X. Model checking of dimension-reduction type for regression. Statist. Sinica (2003) 13:283–96.
Zhu L. X., Ng K. W. Checking the adequacy of a partial linear model. Statist. Sinica (2003) 13:763–81.
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