TY - JOUR
T1 - Nonparametric inferences for additive models
AU - Fan, Jianqing
AU - Jiang, Jiancheng
N1 - Funding Information:
Jianqing Fan is Professor, Department of Operation Research and Financial Engineering, Princeton University, Princeton, NJ 08544, and Professor of Statistics, Chinese University of Hong Kong (E-mail: [email protected]). Jiancheng Jiang is Associate Professor, Department of Probability & Statistics, Peking University, Beijing 100871, China (E-mail: [email protected]). Fan was supported in part by RGC grant CUHK 4262/01P of the HKSAR, National Science Foundation grants DMS-03-55179 and DMS-03-54223, and National Institutes of Health grant R01 HL69720, and Jiang was supported by Chinese National Science Foundation grants 10001004 and 39930160.
PY - 2005/9
Y1 - 2005/9
N2 - Additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques. However, the inferences of the models have not been very well developed, due partially to the complexity of the backfitting estimators. There are few tools available to answer some important and frequently asked questions, such as whether a specific additive component is significant or admits a certain parametric form. In an attempt to address these issues, we extend the generalized likelihood ratio (GLR) tests to additive models, using the backfitting estimator. We demonstrate that under the null models, the newly proposed GLR statistics follow asymptotically rescaled chi-squared distributions, with the scaling constants and the degrees of freedom independent of the nuisance parameters. This demonstrates that the Wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspecified error distributions. We further prove that the GLR tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. In addition, for testing a parametric additive model, we propose a bias corrected method to improve the performance of the GLR. The bias-corrected test is shown to share the Wilks type of property. Simulations are conducted to demonstrate the Wilks phenomenon and the power of the proposed tests. A real example is used to illustrate the performance of the testing approach.
AB - Additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques. However, the inferences of the models have not been very well developed, due partially to the complexity of the backfitting estimators. There are few tools available to answer some important and frequently asked questions, such as whether a specific additive component is significant or admits a certain parametric form. In an attempt to address these issues, we extend the generalized likelihood ratio (GLR) tests to additive models, using the backfitting estimator. We demonstrate that under the null models, the newly proposed GLR statistics follow asymptotically rescaled chi-squared distributions, with the scaling constants and the degrees of freedom independent of the nuisance parameters. This demonstrates that the Wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspecified error distributions. We further prove that the GLR tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. In addition, for testing a parametric additive model, we propose a bias corrected method to improve the performance of the GLR. The bias-corrected test is shown to share the Wilks type of property. Simulations are conducted to demonstrate the Wilks phenomenon and the power of the proposed tests. A real example is used to illustrate the performance of the testing approach.
KW - Additive models
KW - Backfitting algorithm
KW - Generalized likelihood ratio
KW - Local polynomial regression
KW - Wilks phenomenon
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U2 - 10.1198/016214504000001439
DO - 10.1198/016214504000001439
M3 - Review article
AN - SCOPUS:24644452605
SN - 0162-1459
VL - 100
SP - 890
EP - 907
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 471
ER -