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Dimension reduction-based significance testing in nonparametric regression. (English) Zbl 1454.62134
Summary: A dimension reduction-based adaptive-to-model test is proposed for significance of a subset of covariates in the context of a nonparametric regression model. Unlike existing locally smoothing significance tests, the new test behaves like a locally smoothing test as if the number of covariates was just that under the null hypothesis and it can detect local alternative hypotheses distinct from the null hypothesis at the rate that is only related to the number of covariates under the null hypothesis. Thus, the curse of dimensionality is largely alleviated when nonparametric estimation is inevitably required. In the cases where there are many insignificant covariates, the improvement of the new test is very significant over existing locally smoothing tests on the significance level maintenance and power enhancement. Simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed test.
Reviewer: Reviewer (Berlin)

MSC:
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62H15 Hypothesis testing in multivariate analysis
62H10 Multivariate distribution of statistics
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References:
[1] Cook, R. D. (1998)., Regression graphics: ideas for studying regressions through graphics. Wiley, New York. · Zbl 0903.62001
[2] Cook, R. D. and Forzani, L. (2009). Likelihood-based sufficient dimension reduction., Journal of the American Statistical Association. 104 197-208. · Zbl 1388.62041
[3] Cook, R. D. and Weisberg, S. (1991). Discussion of “Sliced inverse regression for dimension reduction,” by K. C. Li., Journal of the American Statistical Association. 86 316-342. · Zbl 1353.62037
[4] Delgado, M. A. and González Manteiga, W. (2001). Significance testing in nonparametric regression based on the bootstrap., The Annals of Statistics. 29 1469-1507. · Zbl 1043.62032
[5] Fan, J. Q. and Hu, T. C. (1992). Bias correction and higher order kernel functions., Statistics & probability letters. 13 235-243. · Zbl 0741.62048
[6] Fan, Y. Q. and Li, Q. (1996). Consistent model specification tests: omitted variables and semiparametric functional forms., Econometrica. 64 865-890. · Zbl 0854.62038
[7] Friendly, M. (2002). Corrgrams: Exploratory displays for correlation matrices., The American Statistician. 56 316-324.
[8] Guo, X., Wang. T. and Zhu, L. X. (2016). Model checking for generalized linear models: a dimension-reduction model-adaptive approach., Journal of the Royal Statistical Society: Series B. 78 1013-1035. · Zbl 1414.62131
[9] Hoaglin, D. C. and Velleman, P. F. (1995). A critical look at some analyses of major league baseball salaries., The American Statistician. 49 277-285.
[10] Lavergne, P. and Vuong, Q. (2000). Nonparametric significance testing., Econometric Theory. 16 576-601. · Zbl 0968.62047
[11] Lavergne, P., Maistre, S. and Patilea, V. (2015). A significance test for covariates in nonparametric regression., Electronic Journal of Statistics. 9 643-678. · Zbl 1309.62076
[12] Li, B. and Wang, S. (2007). On directional regression for dimension reduction., Journal of the American Statistical Association. 102 997-1008. · Zbl 05564427
[13] Li, B., Wen, S. Q. and Zhu, L. X. (2008). On a projective resampling method for dimension reduction with multivariate responses., Journal of the American Statistical Association. 103 1177-1186. · Zbl 1205.62067
[14] Li, K. C. (1991). Sliced inverse regression for dimension reduction., Journal of the American Statistical Association. 86 316-327. · Zbl 0742.62044
[15] Li, Q. (1999). Consistent model specification tests for time series econometric models., Journal of Econometrics. 92 101-147. · Zbl 0929.62054
[16] Powell, J. L., Stock, J. H. and Stoker, T. M. (1989). Semiparametric estimation of index coefficients., Econometrica. 57 1403-1430. · Zbl 0683.62070
[17] Racine, J. (1997). Consistent significance testing for nonparametric regression., Journal of Business & Economic Statistics. 15 369-378.
[18] Racine, J. S., Hart, J. and Li, Q. (2006). Testing the significance of categorical predictor variables in nonparametric regression models., Econometric Reviews. 25 523-544. · Zbl 1106.62046
[19] Robinson, P. M. (1988). Root-N-consistent semiparametric regression., Econometrica. 56 931-954. · Zbl 0647.62100
[20] Stute, W. and Zhu, L. X. (2005). Nonparametric checks for single-index models., The Annals of Statistics. 33 1048-1083. · Zbl 1080.62023
[21] Xia, Q., Xu, W. L. and Zhu, L. X. (2015). Consistently determining the number of factors in multivariate volatility modelling., The Statistica Sinica. 25 1025-1044. · Zbl 1415.62067
[22] Xia, Y. C., Tong, H., Li, W. K. and Zhu, L. X. (2002). An adaptive estimation of dimension reduction space., Journal of the Royal Statistical Society: Series B. 64 363-410. · Zbl 1091.62028
[23] Zheng, J. X. (1996). A consistent test of functional form via nonparametric estimation techniques., Journal of Econometrics. 75 263-289. · Zbl 0865.62030
[24] Zhu, L. P., Zhu, L. X., Ferré, L. and Wang, T. (2010a). Sufficient dimension reduction through discretization-expectation estimation., Biometrika. 97 295-304. · Zbl 1205.62048
[25] Zhu, L. P., Zhu, L. X. and Feng, Z. H. (2010b). Dimension reduction in regressions through cumulative slicing estimation., Journal of the American Statistical Association. 105 1455-1466. · Zbl 1388.62121
[26] Zhu, L. X. and Fang, K. T. (1996). Asymptotics for the kernel estimates of sliced inverse regression., The Annals of Statistics. 24 1053-1067. · Zbl 0864.62027
[27] Zhu, L. X., Miao, B. Q. and Peng, H. (2006). On sliced inverse regression with high dimensional covariates., Journal of the American Statistical Association. 101 630-643. · Zbl 1119.62331
[28] Zhu, L. X. and Ng, K. W. (1995). Asymptotics for sliced inverse regression., Statistica Sinica. 5 727-736. · Zbl 0824.62036
[29] Zhu, X. H., Chen, F., Guo, X. and Zhu, L. X. (2016). Heteroscedasticity testing for regression models: A dimension reduction-based model adaptive approach., Computational Statistics & Data Analysis. 103 263-283. · Zbl 06918228
[30] Zhu, X. H., Guo, X. and Zhu, L. X. (2017a). An adaptive-to-model test for partially parametric single-index models., Statistics and Computing. 27 1193-1204. · Zbl 06737706
[31] Zhu, X. H., Wang, T., Zhao, J. L. and Zhu, L. X. (2017b). Inference for biased transformation models., Computational Statistics & Data Analysis. 109 105-120. · Zbl 06917823
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