×

Nonlinear measurement errors models subject to partial linear additive distortion. (English) Zbl 1404.62042

Summary: We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by partial linear additive models (PLAM) of some observed confounding variables. After approximating the additive nonparametric components in the PLAM via polynomial splines and calibrating the unobserved response and unobserved predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. The resulting estimators are shown to be asymptotically normal. To construct confidence intervals for the parameters of interest, an empirical likelihood-based statistic is proposed to improve the accuracy of the associated normal approximation. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure based on the empirical process is proposed to check whether the parametric regression model is adequate or not. A wild bootstrap procedure is proposed to compute \(p\)-values. Simulation studies are conducted to examine the performance of the estimation and testing procedures. The methods are applied to re-analyze real data from a diabetes study for an illustration.

MSC:

62G08 Nonparametric regression and quantile regression
62J02 General nonlinear regression
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] 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
[2] Chen, R., Liang, H. and Wang, J. (2011). Determination of linear components in additive models. Journal of Nonparametric Statistics23, 367–383. · Zbl 1317.62033
[3] Cui, X., Guo, W., Lin, L. and Zhu, L. (2009). Covariate-adjusted nonlinear regression. The Annals of Statistics37, 1839–1870. · Zbl 1168.62035
[4] Cui, X., Härdle, W. and Zhu, L.-X. (2011). The EFM approach for single-index models. The Annals of Statistics39, 1658–1688. · Zbl 1221.62062
[5] de Boor, C. (2001). A Practical Guide to Splines, Revised ed. Applied Mathematical Sciences27. New York: Springer. · Zbl 0987.65015
[6] Delaigle, A., Hall, P. and Wen-Xin, Z. (2016). Nonparametric covariate-adjusted regression. The Annals of Statistics. To appear. Available at arXiv:1601.02739. · Zbl 1349.62097
[7] Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory22, 1030–1051. · Zbl 1170.62318
[8] Eubank, R. L. and Spiegelman, C. H. (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques. Journal of the American Statistical Association85, 387–392. · Zbl 0702.62037
[9] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association96, 1348–1360. · Zbl 1073.62547
[10] Härdle, W. and Liang, H. (2007). Partially linear models. In Statistical Methods for Biostatistics and Related Fields, 87–103. Berlin: Springer.
[11] Härdle, W., Liang, H. and Gao, J. T. (2000). Partially Linear Models. Heidelberg: Springer.
[12] Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. The Annals of Statistics21, 1926–1947. · Zbl 0795.62036
[13] Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer Series in Statistics. New York: Springer. · Zbl 0886.62043
[14] Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. London: Chapman and Hall. · Zbl 0747.62061
[15] Heckman, N. E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B48, 244–248. · Zbl 0623.62030
[16] Huang, J. (1999). Efficient estimation of the partly linear additive Cox model. The Annals of Statistics27, 1536–1563. · Zbl 0977.62035 · doi:10.1214/aos/1017939141
[17] Huang, J. Z. (2003). Local asymptotics for polynomial spline regression. The Annals of Statistics31, 1600–1635. · Zbl 1042.62035
[18] Kaysen, G. A., Dubin, J. A., Müller, H.-G., Mitch, W. E., Rosales, L. M., Levin, N. W. and the Hemo Study Group (2002). Relationships among inflammation nutrition and physiologic mechanisms establishing albumin levels in hemodialysis patients. Kidney International61, 2240–2249.
[19] Li, F., Lin, L. and Cui, X. (2010). Covariate-adjusted partially linear regression models. Communications in Statistics. Theory and Methods39, 1054–1074. · Zbl 1284.62422
[20] Li, G., Lin, L. and Zhu, L. (2012). Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters. Journal of Multivariate Analysis105, 85–111. · Zbl 1236.62020
[21] Li, Q. (2000). Efficient estimation of additive partially linear models. International Economic Review41, 1073–1092.
[22] Li, X., Du, J., Li, G. and Fan, M. (2014). Variable selection for covariate adjusted regression model. Journal of Systems Science and Complexity27, 1227–1246. · Zbl 1310.62086
[23] Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines. Biometrika95, 415–436. · Zbl 1437.62540
[24] Lian, H. (2012). Empirical likelihood confidence intervals for nonparametric functional data analysis. Journal of Statistical Planning and Inference142, 1669–1677. · Zbl 1238.62057
[25] Liang, H. and Li, R. (2009). Variable selection for partially linear models with measurement errors. Journal of the American Statistical Association104, 234–248. · Zbl 1388.62208
[26] Liang, H., Qin, Y., Zhang, X. and Ruppert, D. (2009a). Empirical likelihood-based inferences for generalized partially linear models. Scandinavian Journal of Statistics. Theory and Applications36, 433–443. · Zbl 1197.62092
[27] Liang, H., Su, H., Thurston, S. W., Meeker, J. D. and Hauser, R. (2009b). Empirical likelihood based inference for additive partial linear measurement error models. Statistics and its Interface2, 83–90. · Zbl 1245.62024
[28] Liang, H., Thurston, S. W., Ruppert, D., Apanasovich, T. and Hauser, R. (2008). Additive partial linear models with measurement errors. Biometrika95, 667–678. · Zbl 1437.62526
[29] Liu, X., Wang, L. and Liang, H. (2011). Estimation and variable selection for semiparametric additive partial linear models. Statistica Sinica21, 1225–1248. · Zbl 1223.62020
[30] Mammen, E. (1993). Bootstrap and wild bootstrap for high-dimensional linear models. The Annals of Statistics21, 255–285. · Zbl 0771.62032
[31] Nguyen, D. V. and Şentürk, D. (2008). Multicovariate-adjusted regression models. Journal of Statistical Computation and Simulation78, 813–827. · Zbl 1431.62167
[32] Opsomer, J. D. and Ruppert, D. (1997). Fitting a bivariate additive model by local polynomial regression. The Annals of Statistics25, 186–211. · Zbl 0869.62026
[33] Owen, A. B. (2001). Empirical Likelihood. London: Chapman and Hall/CRC. · Zbl 0989.62019
[34] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. The Annals of Statistics22, 300–325. · Zbl 0799.62049 · doi:10.1214/aos/1176325370
[35] Şentürk, D. and Müller, H.-G. (2005). Covariate-adjusted regression. Biometrika92, 75–89.
[36] Şentürk, D. and Müller, H.-G. (2006). Inference for covariate adjusted regression via varying coefficient models. The Annals of Statistics34, 654–679.
[37] Şentürk, D. and Müller, H.-G. (2009). Covariate-adjusted generalized linear models. Biometrika96, 357–370.
[38] Şentürk, D. and Nguyen, D. V. (2009). Partial covariate adjusted regression. Journal of Statistical Planning and Inference139, 454–468. · Zbl 1149.62061
[39] Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B50, 413–436. · Zbl 0671.62045
[40] Stone, C. J. (1985). Additive regression and other nonparametric models. The Annals of Statistics13, 689–705. · Zbl 0605.62065 · doi:10.1214/aos/1176349548
[41] Stute, W., González Manteiga, W. and Presedo Quindimil, M. (1998a). Bootstrap approximations in model checks for regression. Journal of the American Statistical Association93, 141–149. · Zbl 0902.62027
[42] Stute, W., González Manteiga, W. and Presedo Quindimil, M. (1998b). Bootstrap approximations in model checks for regression. Journal of the American Statistical Association93, 141–149. · Zbl 0902.62027
[43] Stute, W., Thies, S. and Zhu, L.-X. (1998). Model checks for regression: An innovation process approach. The Annals of Statistics26, 1916–1934. · Zbl 0930.62044 · doi:10.1214/aos/1024691363
[44] Tang, N.-S. and Zhao, P.-Y. (2013a). Empirical likelihood-based inference in nonlinear regression models with missing responses at random. Statistics. A Journal of Theoretical and Applied Statistics47, 1141–1159. · Zbl 1440.62262
[45] Tang, N.-S. and Zhao, P.-Y. (2013b). Empirical likelihood semiparametric nonlinear regression analysis for longitudinal data with responses missing at random. Annals of the Institute of Statistical Mathematics65, 639–665. · Zbl 1273.62091 · doi:10.1007/s10463-012-0387-4
[46] Wang, L., Liu, X., Liang, H. and Carroll, R. J. (2011). Estimation and variable selection for generalized additive partial linear models. The Annals of Statistics39, 1827–1851. · Zbl 1227.62053
[47] Wang, X., Li, G. and Lin, L. (2011). Empirical likelihood inference for semi-parametric varying-coefficient partially linear EV models. Metrika. International Journal for Theoretical and Applied Statistics73, 171–185. · Zbl 1206.62080
[48] Wei, Z. and Zhu, L. (2010). Evaluation of value at risk: An empirical likelihood approach. Statistica Sinica20, 455–468. · Zbl 1180.62153
[49] Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics14, 1261–1350. · Zbl 0618.62072 · doi:10.1214/aos/1176350142
[50] Xia, Y., Li, W. K., Tong, H. and Zhang, D. (2004). A goodness-of-fit test for single-index models. Statistica Sinica14, 1–39. · Zbl 1040.62034
[51] Zhang, J., Feng, S., Li, G. and Lian, H. (2011). Empirical likelihood inference for partially linear panel data models with fixed effects. Economics Letters113, 165–167. · Zbl 1238.62049
[52] Zhang, J., Feng, Z. and Zhou, B. (2014). A revisit to correlation analysis for distortion measurement error data. Journal of Multivariate Analysis124, 116–129. · Zbl 1278.62065
[53] Zhang, J., Gai, Y. and Wu, P. (2013). Estimation in linear regression models with measurement errors subject to single-indexed distortion. Computational Statistics & Data Analysis59, 103–120.
[54] Zhang, J., Li, G. and Feng, Z. (2015). Checking the adequacy for a distortion errors-in-variables parametric regression model. Computational Statistics & Data Analysis83, 52–64.
[55] Zhang, J., Yu, Y., Zhou, B. and Liang, H. (2014). Nonlinear measurement errors models subject to additive distortion. Journal of Statistical Planning and Inference150, 49–65. · Zbl 1287.62012 · doi:10.1016/j.jspi.2014.03.005
[56] Zhang, J., Yu, Y., Zhu, L. and Liang, H. (2013). Partial linear single index models with distortion measurement errors. Annals of the Institute of Statistical Mathematics65, 237–267. · Zbl 1440.62141 · doi:10.1007/s10463-012-0371-z
[57] Zhang, J., Zhou, N., Sun, Z., Li, G. and Wei, Z. (2016). Statistical inference on restricted partial linear regression models with partial distortion measurement errors. Statistica Neerlandica. Journal of the Netherlands Society for Statistics and Operations Research. 70, 304–331. Available at DOI:10.1111/stan.12089.
[58] Zhang, J., Zhu, L. and Liang, H. (2012). Nonlinear models with measurement errors subject to single-indexed distortion. Journal of Multivariate Analysis112, 1–23. · Zbl 1274.62304 · doi:10.1016/j.jmva.2012.05.012
[59] Zhang, X. and Liang, H. (2011). Focused information criterion and model averaging for generalized additive partial linear models. The Annals of Statistics39, 194–200.
[60] Zhu, L., Lin, L., Cui, X. and Li, G. (2010). Bias-corrected empirical likelihood in a multi-link semiparametric model. Journal of Multivariate Analysis101, 850–868. · Zbl 1181.62039 · doi:10.1016/j.jmva.2009.08.009
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.