×

zbMATH — the first resource for mathematics

Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models. (English) Zbl 07135457
Summary: In this paper we consider partial linear regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration, which avoids to use the nonzero expectation conditions imposed on the variables. With these calibrated variables, a profile least squares estimator is obtained, associated with its normal approximation based and empirical likelihood based confidence intervals. For the hypothesis testing on parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Lastly, a score-type test statistic is then proposed for checking the validity of partial linear models. We derive asymptotic distribution of the proposed test statistic. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives, which converge to the null hypothesis at a parametric rate. Simulation studies demonstrate the performance of our proposed procedure and a real example is analyzed as illustrate its practical usage.
MSC:
62 Statistics
Software:
KernSmooth
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Carroll, R. J.; Ruppert, D.; Stefanski, L. A.; Crainiceanu, C. M., Nonlinear Measurement Error Models, A Modern Perspective (2006), Chapman and Hall: Chapman and Hall New York · Zbl 1119.62063
[2] Cook, R. D.; Weisberg, S., Residuals and Influence in Regression (1982), Chapman and Hall: Chapman and Hall New York · Zbl 0564.62054
[3] Cui, X.; Guo, W.; Lin, L.; Zhu, L., Covariate-adjusted nonlinear regression, Ann. Statist., 37, 1839-1870 (2009) · Zbl 1168.62035
[4] Delaigle, A.; Hall, P.; Zhou, W.-X., Nonparametric covariate-adjusted regression, Ann. Statist., 44, 5, 2190-2220 (2016) · Zbl 1349.62097
[5] Fan, J.; Peng, H., Nonconcave penalized likelihood with a diverging number of parameters, Ann. Statist., 32, 3, 928-961 (2004) · Zbl 1092.62031
[6] Fuller, W. A., Measurement Error Models, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, xxiv+440 (1987), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0800.62413
[7] Härdle, W.; Liang, H.; Gao, J. T., Partially Linear Models (2000), Springer Physica: Springer Physica Heidelberg
[8] Hwang, J. T., Multiplicative errors-in-variables models with applications to recent data released by the U.S. department of energy, J. Amer. Statist. Assoc., 81, 395, 680-688 (1986) · Zbl 0621.62072
[9] Li, F.; Lin, L.; Cui, X., Covariate-adjusted partially linear regression models, Commun. Statist. Theory Methods, 39, 6, 1054-1074 (2010) · Zbl 1284.62422
[10] Li, F.; Lin, L.; Lu, Y.; Feng, S., An adaptive estimation for covariate-adjusted nonparametric regression model, Statist. Papers (2018)
[11] Li, G.; Lin, L.; Zhu, L., Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters, J. Multivariate Anal., 105, 85-111 (2012) · Zbl 1236.62020
[12] Li, F.; Lu, Y., Lasso-type estimation for covariate-adjusted linear model, J. Appl. Stat., 45, 1, 26-42 (2018)
[13] Li, G.; Zhang, J.; Feng, S., Modern Measurement Error Models (2016), Science Press: Science Press Beijing
[14] Lian, H., Empirical likelihood confidence intervals for nonparametric functional data analysis, J. Statist. Plann. Inference, 142, 7, 1669-1677 (2012) · Zbl 1238.62057
[15] Lian, H.; Liang, H.; Wang, L., Generalized additive partial linear models for clustered data with diverging number of covariates using gee, Statist. Sinica, 24, 1, 173-196 (2014) · Zbl 1285.62080
[16] Liang, H.; Härdle, W.; Carroll, R. J., Estimation in a semiparametric partially linear errors-in-variables model, Ann. Statist., 27, 5, 1519-1535 (1999) · Zbl 0977.62036
[17] Liang, H.; Liu, X.; Li, R.; Tsai, C. L., Estimation and testing for partially linear single-index models, Ann. Statist., 38, 3811-3836 (2010) · Zbl 1204.62068
[18] Liang, H.; Qin, Y.; Zhang, X.; Ruppert, D., Empirical likelihood-based inferences for generalized partially linear models, Scand. J. Statist. Theory Appl., 36, 3, 433-443 (2009) · Zbl 1197.62092
[19] Liu, J.; Lou, L.; Li, R., Variable selection for partially linear models via partial correlation, J. Multivariate Anal., 167, 418-434 (2018) · Zbl 1395.62203
[20] Nguyen, D. V.; Şentürk, D., Distortion diagnostics for covariate-adjusted regression: Graphical techniques based on local linear modeling, J. Data Sci., 5, 471-490 (2007)
[21] Nguyen, D. V.; Şentürk, D., Multicovariate-adjusted regression models, J. Stat. Comput. Simul., 78, 813-827 (2008) · Zbl 1431.62167
[22] Nguyen, D. V.; Şentürk, D.; Carroll, R. J., Covariate-adjusted linear mixed effects model with an application to longitudinal data, J. Nonparametr. Stat., 20, 459-481 (2008) · Zbl 1145.62032
[23] Owen, A. B., Empirical Likelihood (2001), Chapman and Hall/CRC: Chapman and Hall/CRC London · Zbl 0989.62019
[24] Şentürk, D.; Müller, H.-G., Covariate adjusted correlation analysis via varying coefficient models, Scand. J. Statist. Theory Appl., 32, 3, 365-383 (2005) · Zbl 1089.62068
[25] Şentürk, D.; Müller, H.-G., Inference for covariate adjusted regression via varying coefficient models, Ann. Statist., 34, 654-679 (2006) · Zbl 1095.62045
[26] Şentürk, D.; Nguyen, D. V., Estimation in covariate-adjusted regression, Comput. Statist. Data Anal., 50, 11, 3294-3310 (2006) · Zbl 1445.62083
[27] Şentürk, D.; Nguyen, D. V., Partial covariate adjusted regression, J. Statist. Plann. Inference, 139, 2, 454-468 (2009) · Zbl 1149.62061
[28] Stute, W.; Zhu, L.-X., Nonparametric checks for single-index models, Ann. Statist., 33, 1048-1083 (2005) · Zbl 1080.62023
[29] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B, 58, 267-288 (1996) · Zbl 0850.62538
[30] Wand, M. P.; Jones, M. C., Kernel Smoothing, Monographs on Statistics and Applied Probability, xii+212 (1995), Chapman and Hall, Ltd.: Chapman and Hall, Ltd. London · Zbl 0854.62043
[31] Wang, L.; Xue, L.; Qu, A.; Liang, H., Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates, Ann. Statist., 42, 2, 592-624 (2014) · Zbl 1309.62077
[32] Xie, C.; Zhu, L., A goodness-of-fit test for variable-adjusted models, Comput. Statist. Data Anal., 138, 27-48 (2019) · Zbl 07060669
[33] Yang, Y.; Tong, T.; Li, G., Simex estimation for single-index model with covariate measurement error, AStA Adv. Stat. Anal., 103, 1, 137-161 (2019) · Zbl 1427.62022
[34] Zhang, J.; Zhou, N.; Sun, Z.; Li, G.; Wei, Z., Statistical inference on restricted partial linear regression models with partial distortion measurement errors, Stat. Neerl., 70, 4, 304-331 (2016)
[35] Zhao, J.; Xie, C., A nonparametric test for covariate-adjusted models, Statist. Probab. Lett., 133, 65-70 (2018) · Zbl 1439.62113
[36] Zhu, L.; Cui, H., Testing the adequacy for a general linear errors-in-variables model, Statist. Sinica, 15, 4, 1049-1068 (2005) · Zbl 1086.62026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.