×

Statistical inference on restricted linear regression models with partial distortion measurement errors. (English) Zbl 1381.62233

Summary: We consider statistical inference for linear regression models when some variables are distorted with errors by some unknown functions of commonly observable confounding variables. The proposed estimation procedure is designed to accommodate undistorted as well as distorted variables. To test a hypothesis on the parametric components, a restricted least squares estimator is proposed for unknown parameters under some restricted conditions. Asymptotic properties for the estimators are established. A test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we also obtain the asymptotic properties of the test statistic. A wild bootstrap procedure is proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for an illustration.

MSC:

62J05 Linear regression; mixed models
62F12 Asymptotic properties of parametric estimators
62F40 Bootstrap, jackknife and other resampling methods
62G05 Nonparametric estimation
62H15 Hypothesis testing in multivariate analysis
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association 92 , 477-489. · Zbl 0890.62053 · doi:10.2307/2965697
[2] Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Nonlinear Measurement Error Models. A Modern Perspective , 2nd ed. New York: Chapman and Hall. · Zbl 1119.62063 · doi:10.1201/9781420010138
[3] Cui, X., Guo, W., Lin, L. and Zhu, L. (2009). Covariate-adjusted nonlinear regression. The Annals of Mathematical Statistics 37 , 1839-1870. · Zbl 1168.62035 · doi:10.1214/08-AOS627
[4] Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory 22 , 1030-1051. · Zbl 1170.62318 · doi:10.1017/S0266466606060506
[5] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . London: Chapman and Hall. · Zbl 0873.62037
[6] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96 , 1348-1360. · Zbl 1073.62547 · doi:10.1198/016214501753382273
[7] Fan, J. and Zhang, J. (2000). Two-step estimation of functional linear models with applications to longitudinal data. Journal of the Royal Statistical Society, Series B, Statistical Methodology 62 , 303-322.
[8] Kaysen, G. A., Dubin, J. A., Müller, H.-G., Mitch, W. E., Rosales, L. M. and Nathan W. Levin the Hemo Study Group (2002). Relationships among inflammation nutrition and physiologic mechanisms establishing albumin levels in hemodialysis patients. Kidney International 61 , 2240-2249.
[9] Kneip, A., Simar, L. and Van Keilegom, I. (2015). Frontier estimation in the presence of measurement error with unknown variance. Journal of Econometrics 184 , 379-393. · Zbl 1331.62478 · doi:10.1016/j.jeconom.2014.09.012
[10] Li, F., Lin, L. and Cui, X. (2010). Covariate-adjusted partially linear regression models. Communications in Statistics. Theory and Methods 39 , 1054-1074. · Zbl 1284.62422 · doi:10.1080/03610920902846539
[11] Li, G. and Xue, L. (2008). Empirical likelihood confidence region for the parameter in a partially linear errors-in-variables model. Communications in Statistics. Theory and Methods 37 , 1552-1564. · Zbl 1189.62083 · doi:10.1080/03610920801893913
[12] Li, X., Du, J., Li, G. and Fan, M. (2014). Variable selection for covariate adjusted regression model. Journal of Systems Science and Complexity 27 , 1227-1246. · Zbl 1310.62086 · doi:10.1007/s11424-014-2276-9
[13] Liang, H. and Li, R. (2009). Variable selection for partially linear models with measurement errors. Journal of the American Statistical Association 104 , 234-248. · Zbl 1388.62208
[14] Mack, Y. P. and Silverman, B. W. (1982). Weak and strong uniform consistency of kernel regression estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61 , 405-415. · Zbl 0495.62046 · doi:10.1007/BF00539840
[15] Nguyen, D. V., Şentürk, D. and Carroll, R. J. (2008). Covariate-adjusted linear mixed effects model with an application to longitudinal data. Journal of Nonparametric Statistics 20 , 459-481. · Zbl 1145.62032 · doi:10.1080/10485250802226435
[16] Saleh, A. E. and Shalabh (2014). A ridge regression estimation approach to the measurement error model. Journal of Multivariate Analysis 123 , 68-84. · Zbl 1360.62402 · doi:10.1016/j.jmva.2013.08.014
[17] Şentürk, D. and Müller, H.-G. (2005). Covariate adjusted correlation analysis via varying coefficient models. Scandinavian Journal of Statistics 32 , 365-383. · Zbl 1089.62068 · doi:10.1111/j.1467-9469.2005.00450.x
[18] Şentürk, D. and Müller, H.-G. (2006). Inference for covariate adjusted regression via varying coefficient models. The Annals of Mathematical Statistics 34 , 654-679. · Zbl 1095.62045 · doi:10.1214/009053606000000083
[19] Şentürk, D. and Nguyen, D. V. (2006). Estimation in covariate-adjusted regression. Computational Statistics & Data Analysis 50 , 3294-3310. · Zbl 1445.62083
[20] Şentürk, D. and Nguyen, D. V. (2009). Partial covariate adjusted regression. Journal of Statistical Planning and Inference 139 , 454-468. · Zbl 1149.62061 · doi:10.1016/j.jspi.2008.04.030
[21] Sheil, J. and O’Muircheartaigh, I. (1977). Algorithm as106: The distribution of non-negative quadratic forms in normal variables. Applied Statistics 26 , 92-98.
[22] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. London: Chapman and Hall. · Zbl 0617.62042
[23] Stefanski, L., Wu, Y. and White, K. (2014). Variable selection in nonparametric classification via measurement error model selection likelihoods. Journal of the American Statistical Association 109 , 574-589. · Zbl 1367.62219
[24] Stute, W., González Manteiga, W. and Presedo Quindimil, M. (1998). Bootstrap approximations in model checks for regression. Journal of the American Statistical Association 93 , 141-149. · Zbl 0902.62027 · doi:10.2307/2669611
[25] Wei, C. and Wang, Q. (2012). Statistical inference on restricted partially linear additive errors-in-variables models. Test 21 , 757-774. · Zbl 1284.62286 · doi:10.1007/s11749-011-0279-6
[26] Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Mathematical Statistics 14 , 1261-1350. · Zbl 0618.62072 · doi:10.1214/aos/1176350142
[27] Xu, W. and Guo, X. (2013). Nonparametric checks for a varying coefficient model with missing response at random. Metrika 76 , 459-482. · Zbl 1353.62046 · doi:10.1007/s00184-012-0399-3
[28] Xu, W. and Zhu, L. (2013). Testing the adequacy of varying coefficient models with missing responses at random. Metrika 76 , 53-69. · Zbl 1256.62029 · doi:10.1007/s00184-011-0375-3
[29] Xu, W. and Zhu, L. (2015). Nonparametric check for partial linear errors-in-covariables models with validation data. Annals of the Institute of Statistical Mathematics 67 , 793-815. · Zbl 1440.62139
[30] Yang, Y., Li, G. and Peng, H. (2014). Empirical likelihood of varying coefficient errors-in-variables models with longitudinal data. Journal of Multivariate Analysis 127 , 1-18. · Zbl 1293.62084 · doi:10.1016/j.jmva.2014.02.004
[31] Zhang, J., Feng, Z. and Zhou, B. (2014a). A revisit to correlation analysis for distortion measurement error data. Journal of Multivariate Analysis 124 , 116-129. · Zbl 1278.62065 · doi:10.1016/j.jmva.2013.10.004
[32] Zhang, J., Gai, Y. and Wu, P. (2013a). Estimation in linear regression models with measurement errors subject to single-indexed distortion. Computational Statistics & Data Analysis 59 , 103-120.
[33] Zhang, J., Li, G. and Feng, Z. (2015). Checking the adequacy for a distortion errors-in-variables parametric regression model. Computational Statistics & Data Analysis 83 , 52-64. · Zbl 1507.62205
[34] Zhang, J., Wang, X., Yu, Y. and Gai, Y. (2014b). Estimation and variable selection in partial linear single index models with error-prone linear covariates. Statistics. A Journal of Theoretical and Applied Statistics 48 , 1048-1070. · Zbl 1367.62131
[35] Zhang, J., Yu, Y., Zhou, B. and Liang, H. (2014c). Nonlinear measurement errors models subject to additive distortion. Journal of Statistical Planning and Inference 150 , 49-65. · Zbl 1287.62012 · doi:10.1016/j.jspi.2014.03.005
[36] Zhang, J., Yu, Y., Zhu, L. and Liang, H. (2013b). Partial linear single index models with distortion measurement errors. Annals of the Institute of Statistical Mathematics 65 , 237-267. · Zbl 1440.62141
[37] Zhang, J., Zhu, L. and Liang, H. (2012a). Nonlinear models with measurement errors subject to single-indexed distortion. Journal of Multivariate Analysis 112 , 1-23. · Zbl 1274.62304 · doi:10.1016/j.jmva.2012.05.012
[38] Zhang, J., Zhu, L. and Zhu, L. (2012b). On a dimension reduction regression with covariate adjustment. Journal of Multivariate Analysis 104 , 39-55. · Zbl 1231.62076 · doi:10.1016/j.jmva.2011.06.004
[39] Zhang, W., Li, G. and Xue, L. (2011). Profile inference on partially linear varying-coefficient errors-in-variables models under restricted condition. Computational Statistics & Data Analysis 55 , 3027-3040. · Zbl 1218.62038 · doi:10.1016/j.csda.2011.05.012
[40] Zhou, Y. and Liang, H. (2009). Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates. The Annals of Mathematical Statistics 37 , 427-458. · Zbl 1156.62036 · doi:10.1214/07-AOS561
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.