A goodness-of-fit test for variable-adjusted models. (English) Zbl 1507.62189
Summary: This research provides a projection-based test to check parametric single-index regression structure in variable-adjusted models. An adaptive-to-model strategy is employed, which makes the proposed test work better on the significance level maintenance and more powerful than existing tests. With mild conditions, the proposed test asymptotically behaves like a test that is for classical regression setup without distortion errors in observations. Numerical studies with simulated and real data are conducted to examine the performance of the test in finite sample scenarios.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62G08 | Nonparametric regression and quantile regression |
References:
| [1] | Carroll, R. J.; Ruppert, D.; Crainiceanu, C. M.; Stefanski, L. A., Measurement Error in Nonlinear Models: A Modern Perspective (2006), Chapman and Hall/CRC · Zbl 1119.62063 |
| [2] | Cheng, C.-L.; Van Ness, J. W., Statistical Regression with Measurement Error (1999), Arnold, London · Zbl 0947.62046 |
| [3] | Cook, R. D., Regression Graphics: Ideas for Studying Regressions Through Graphics, Vol. 482 (2009), John Wiley & Sons |
| [4] | Cook, R. D.; Li, B., Dimension reduction for conditional mean in regression, Ann. Statist., 30, 2, 455-474 (2002) · Zbl 1012.62035 |
| [5] | Cook, R. D.; Weisberg, S., Comment, J. Amer. Statist. Assoc., 86, 414, 328-332 (1991) · Zbl 1353.62037 |
| [6] | Cui, X.; Guo, W.; Lin, L.; Zhu, L., Covariate-adjusted nonlinear regression, Ann. Statist., 37, 4, 1839-1870 (2009) · Zbl 1168.62035 |
| [7] | Delaigle, A.; Hall, P.; Zhou, W.-X., Nonparametric covariate-adjusted regression, Ann. Statist., 44, 5, 2190-2220 (2016) · Zbl 1349.62097 |
| [8] | Delgado, M. A.; Escanciano, J. C., Distribution-free tests of conditional moment inequalities, J. Statist. Plann. Inference, 173, 99-108 (2016) · Zbl 1334.62078 |
| [9] | Escanciano, J. C., A consistent diagnostic test for regression models using projections, Econometric Theory, 22, 6, 1030-1051 (2006) · Zbl 1170.62318 |
| [10] | Fuller, W. A., Measurement Error Models, Vol. 305 (2009), John Wiley & Sons |
| [11] | Guo, X.; Wang, T.; Zhu, L., Model checking for parametric single-index models: a dimension reduction model-adaptive approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 78, 5, 1013-1035 (2016) · Zbl 1414.62131 |
| [12] | Harrison, D.; Rubinfeld, D. L., Hedonic housing prices and the demand for clean air, J. Environ. Econ. Manag., 5, 1, 81-102 (1978) · Zbl 0375.90023 |
| [13] | Huber, P. J., Projection pursuit, Ann. Stat., 435-475 (1985) · Zbl 0595.62059 |
| [14] | Lavergne, P.; Patilea, V., Breaking the curse of dimensionality in nonparametric testing, J. Econometrics, 143, 1, 103-122 (2008) · Zbl 1418.62199 |
| [15] | Lavergne, P.; Patilea, V., One for all and all for one: regression checks with many regressors, J. Bus. Econ. Stat., 30, 1, 41-52 (2012) |
| [16] | Li, K.-C., Sliced inverse regression for dimension reduction, J. Amer. Statist. Assoc., 86, 414, 316-327 (1991) · Zbl 0742.62044 |
| [17] | Li, B.; Wang, S., On directional regression for dimension reduction, J. Amer. Statist. Assoc., 102, 479, 997-1008 (2007) · Zbl 1469.62300 |
| [18] | Li, B.; Wen, S.; Zhu, L., On a projective resampling method for dimension reduction with multivariate responses, J. Amer. Statist. Assoc., 103, 483, 1177-1186 (2008) · Zbl 1205.62067 |
| [19] | Li, B.; Zha, H.; Chiaromonte, F., Contour regression: a general approach to dimension reduction, Ann. Statist., 33, 4, 1580-1616 (2005) · Zbl 1078.62033 |
| [20] | Mammen, E.; Rothe, C.; Schienle, M., Nonparametric regression with nonparametrically generated covariates, Ann. Statist., 40, 2, 1132-1170 (2012) · Zbl 1274.62294 |
| [21] | Nguyen, D.; Şentürk, D., Multicovariate-adjusted regression models, J. Stat. Comput. Simul., 78, 9, 813-827 (2008) · Zbl 1431.62167 |
| [22] | Şentürk, D.; Müller, H.-G., Covariate adjusted correlation analysis via varying coefficient models, Scand. J. Stat., 32, 3, 365-383 (2005) · Zbl 1089.62068 |
| [23] | Şentürk, D.; Müller, H.-G., Covariate-adjusted regression, Biometrika, 92, 1, 75-89 (2005) · Zbl 1068.62082 |
| [24] | Stute, W.; Xu, W.; Zhu, L., Model diagnosis for parametric regression in high-dimensional spaces, Biometrika, 95, 2, 451-467 (2008) · Zbl 1437.62614 |
| [25] | Xia, Y.; Tong, H.; Li, W.; Zhu, L.-X., An adaptive estimation of dimension reduction space, J. R. Stat. Soc. Ser. B Stat. Methodol., 64, 3, 363-410 (2002) · Zbl 1091.62028 |
| [26] | Zhang, J.; Li, G.; Feng, Z., Checking the adequacy for a distortion errors-in-variables parametric regression model, Comput. Statist. Data Anal., 83, 52-64 (2015) · Zbl 1507.62205 |
| [27] | Zhang, J.; Yu, Y.; Zhu, L.-X.; Liang, H., Partial linear single index models with distortion measurement errors, Ann. Inst. Statist. Math., 65, 2, 237-267 (2013) · Zbl 1440.62141 |
| [28] | Zhang, J.; Zhu, L.-X.; Liang, H., Nonlinear models with measurement errors subject to single-indexed distortion, J. Multivariate Anal., 112, 1-23 (2012) · Zbl 1274.62304 |
| [29] | Zhang, J.; Zhu, L.-P.; Zhu, L.-X., On a dimension reduction regression with covariate adjustment, J. Multivariate Anal., 104, 1, 39-55 (2012) · Zbl 1231.62076 |
| [30] | Zhao, J.; Xie, C., A nonparametric test for covariate-adjusted models, Statist. Probab. Lett., 133, 65-70 (2018) · Zbl 1439.62113 |
| [31] | Zheng, J. X., A consistent test of functional form via nonparametric estimation techniques, J. Econometrics, 75, 2, 263-289 (1996) · Zbl 0865.62030 |
| [32] | Zhu, L.-X.; Fang, K.-T., Asymptotics for kernel estimate of sliced inverse regression, Ann. Statist., 24, 3, 1053-1068 (1996) · Zbl 0864.62027 |
| [33] | Zhu, L.; Li, R., Dimension-reduction type test for linearity of a stochastic regression model, Acta Math. Appl. Sin., 14, 2, 165-175 (1998) · Zbl 0927.62044 |
| [34] | Zhu, L.; Miao, B.; Peng, H., On sliced inverse regression with high-dimensional covariates, J. Amer. Statist. Assoc., 101, 474, 630-643 (2006) · Zbl 1119.62331 |
| [35] | Zhu, L.; Wang, T.; Zhu, L.; Ferré, L., Sufficient dimension reduction through discretization-expectation estimation, Biometrika, 97, 2, 295-304 (2010) · Zbl 1205.62048 |
| [36] | Zhu, L.-P.; Zhu, L.-X.; Feng, Z.-H., Dimension reduction in regressions through cumulative slicing estimation, J. Amer. Statist. Assoc., 105, 492, 1455-1466 (2010) · Zbl 1388.62121 |
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