## Brazilian Journal of Probability and Statistics

### Nonlinear measurement errors models subject to partial linear additive distortion

#### Abstract

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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 1 (2018), 86-116.

Dates
Accepted: August 2016
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1520046136

Digital Object Identifier
doi:10.1214/16-BJPS333

Mathematical Reviews number (MathSciNet)
MR3770865

Zentralblatt MATH identifier
06973950

#### Citation

Zhang, Jun; Zhou, Nanguang; Chen, Qian; Chu, Tianyue. Nonlinear measurement errors models subject to partial linear additive distortion. Braz. J. Probab. Stat. 32 (2018), no. 1, 86--116. doi:10.1214/16-BJPS333. https://projecteuclid.org/euclid.bjps/1520046136

#### References

• 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.
• Chen, R., Liang, H. and Wang, J. (2011). Determination of linear components in additive models. Journal of Nonparametric Statistics 23, 367–383.
• Cui, X., Guo, W., Lin, L. and Zhu, L. (2009). Covariate-adjusted nonlinear regression. The Annals of Statistics 37, 1839–1870.
• Cui, X., Härdle, W. and Zhu, L.-X. (2011). The EFM approach for single-index models. The Annals of Statistics 39, 1658–1688.
• de Boor, C. (2001). A Practical Guide to Splines, Revised ed. Applied Mathematical Sciences 27. New York: Springer.
• Delaigle, A., Hall, P. and Wen-Xin, Z. (2016). Nonparametric covariate-adjusted regression. The Annals of Statistics. To appear. Available at arXiv:1601.02739.
• Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory 22, 1030–1051.
• 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 Association 85, 387–392.
• 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.
• Härdle, W. and Liang, H. (2007). Partially linear models. In Statistical Methods for Biostatistics and Related Fields, 87–103. Berlin: Springer.
• Härdle, W., Liang, H. and Gao, J. T. (2000). Partially Linear Models. Heidelberg: Springer.
• Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. The Annals of Statistics 21, 1926–1947.
• Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer Series in Statistics. New York: Springer.
• Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. London: Chapman and Hall.
• Heckman, N. E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B 48, 244–248.
• Huang, J. (1999). Efficient estimation of the partly linear additive Cox model. The Annals of Statistics 27, 1536–1563.
• Huang, J. Z. (2003). Local asymptotics for polynomial spline regression. The Annals of Statistics 31, 1600–1635.
• 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 International 61, 2240–2249.
• Li, F., Lin, L. and Cui, X. (2010). Covariate-adjusted partially linear regression models. Communications in Statistics. Theory and Methods 39, 1054–1074.
• 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 Analysis 105, 85–111.
• Li, Q. (2000). Efficient estimation of additive partially linear models. International Economic Review 41, 1073–1092.
• 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.
• Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines. Biometrika 95, 415–436.
• Lian, H. (2012). Empirical likelihood confidence intervals for nonparametric functional data analysis. Journal of Statistical Planning and Inference 142, 1669–1677.
• Liang, H. and Li, R. (2009). Variable selection for partially linear models with measurement errors. Journal of the American Statistical Association 104, 234–248.
• 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 Applications 36, 433–443.
• 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 Interface 2, 83–90.
• Liang, H., Thurston, S. W., Ruppert, D., Apanasovich, T. and Hauser, R. (2008). Additive partial linear models with measurement errors. Biometrika 95, 667–678.
• Liu, X., Wang, L. and Liang, H. (2011). Estimation and variable selection for semiparametric additive partial linear models. Statistica Sinica 21, 1225–1248.
• Mammen, E. (1993). Bootstrap and wild bootstrap for high-dimensional linear models. The Annals of Statistics 21, 255–285.
• Nguyen, D. V. and Şentürk, D. (2008). Multicovariate-adjusted regression models. Journal of Statistical Computation and Simulation 78, 813–827.
• Opsomer, J. D. and Ruppert, D. (1997). Fitting a bivariate additive model by local polynomial regression. The Annals of Statistics 25, 186–211.
• Owen, A. B. (2001). Empirical Likelihood. London: Chapman and Hall/CRC.
• Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. The Annals of Statistics 22, 300–325.
• Şentürk, D. and Müller, H.-G. (2005). Covariate-adjusted regression. Biometrika 92, 75–89.
• Şentürk, D. and Müller, H.-G. (2006). Inference for covariate adjusted regression via varying coefficient models. The Annals of Statistics 34, 654–679.
• Şentürk, D. and Müller, H.-G. (2009). Covariate-adjusted generalized linear models. Biometrika 96, 357–370.
• Şentürk, D. and Nguyen, D. V. (2009). Partial covariate adjusted regression. Journal of Statistical Planning and Inference 139, 454–468.
• Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B 50, 413–436.
• Stone, C. J. (1985). Additive regression and other nonparametric models. The Annals of Statistics 13, 689–705.
• Stute, W., González Manteiga, W. and Presedo Quindimil, M. (1998a). Bootstrap approximations in model checks for regression. Journal of the American Statistical Association 93, 141–149.
• Stute, W., González Manteiga, W. and Presedo Quindimil, M. (1998b). Bootstrap approximations in model checks for regression. Journal of the American Statistical Association 93, 141–149.
• Stute, W., Thies, S. and Zhu, L.-X. (1998). Model checks for regression: An innovation process approach. The Annals of Statistics 26, 1916–1934.
• 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 Statistics 47, 1141–1159.
• 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 Mathematics 65, 639–665.
• Wang, L., Liu, X., Liang, H. and Carroll, R. J. (2011). Estimation and variable selection for generalized additive partial linear models. The Annals of Statistics 39, 1827–1851.
• 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 Statistics 73, 171–185.
• Wei, Z. and Zhu, L. (2010). Evaluation of value at risk: An empirical likelihood approach. Statistica Sinica 20, 455–468.
• Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics 14, 1261–1350.
• Xia, Y., Li, W. K., Tong, H. and Zhang, D. (2004). A goodness-of-fit test for single-index models. Statistica Sinica 14, 1–39.
• Zhang, J., Feng, S., Li, G. and Lian, H. (2011). Empirical likelihood inference for partially linear panel data models with fixed effects. Economics Letters 113, 165–167.
• Zhang, J., Feng, Z. and Zhou, B. (2014). A revisit to correlation analysis for distortion measurement error data. Journal of Multivariate Analysis 124, 116–129.
• Zhang, J., Gai, Y. and Wu, P. (2013). Estimation in linear regression models with measurement errors subject to single-indexed distortion. Computational Statistics & Data Analysis 59, 103–120.
• 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.
• Zhang, J., Yu, Y., Zhou, B. and Liang, H. (2014). Nonlinear measurement errors models subject to additive distortion. Journal of Statistical Planning and Inference 150, 49–65.
• 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 Mathematics 65, 237–267.
• 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.
• Zhang, J., Zhu, L. and Liang, H. (2012). Nonlinear models with measurement errors subject to single-indexed distortion. Journal of Multivariate Analysis 112, 1–23.
• Zhang, X. and Liang, H. (2011). Focused information criterion and model averaging for generalized additive partial linear models. The Annals of Statistics 39, 194–200.
• Zhu, L., Lin, L., Cui, X. and Li, G. (2010). Bias-corrected empirical likelihood in a multi-link semiparametric model. Journal of Multivariate Analysis 101, 850–868.