×

zbMATH — the first resource for mathematics

Covariate-adjusted nonlinear regression. (English) Zbl 1168.62035
Summary: We propose a covariate-adjusted nonlinear regression model. In this model, both the response and predictors can only be observed after being distorted by some multiplicative factors. Because of nonlinearity, existing methods for the linear setting cannot be directly employed. To attack this problem, we propose estimating the distorting functions by nonparametrically regressing the predictors and responses on the distorting covariates; then nonlinear least squares estimators for the parameters are obtained using the estimated responses and predictors. Root \(n\)-consistency and asymptotic normality are established. However, the limiting variance has a very complex structure with several unknown components, and confidence regions based on normal approximations are not efficient. Empirical likelihood-based confidence regions are proposed, and their accuracy is also verified due to its self-scale invariance.
Furthermore, unlike the common results derived from the profile methods, even when plug-in estimates are used for the infinite-dimensional nuisance parameters (distorting functions), the limit of the empirical likelihood ratios is still chi-squared distributed. This property eases the construction of the empirical likelihood-based confidence regions. A simulation study is carried out to assess the finite sample performance of the proposed estimators and confidence regions. We apply our method to study the relationship between glomerular filtration rate and serum creatinine.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62J02 General nonlinear regression
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Andrew, D. R., Timothy, S. L., Erik, J. B., Jeff, M. S., Steven, J. J. and Fernando, G. C. (2004). Using serum creatinine to estimate glomerular filtration rate: Accuracy in good health and in chronic kidney disease. Ann. Intern. Med. 141 929-937.
[2] Eagleson, G. K. and Müller, H. G. (1997). Transformations for smooth regression models with multiplicative errors. J. Roy. Statist. Soc. Ser. B 59 173-189. · Zbl 0889.62030
[3] Fan, J. Q., Lin, H. Z. and Zhou, Y. (2006). Local partial-likelihood for lifetime data. Ann. Statist. 34 290-325. · Zbl 1091.62099 · doi:10.1214/009053605000000796
[4] Fan, J. and Zhang, J. (2000). Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 303-322. JSTOR: · Zbl 04558573 · doi:10.1111/1467-9868.00233 · links.jstor.org
[5] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Int. Statist. Rev. 58 109-127. · Zbl 0716.62003 · doi:10.2307/1403462
[6] Härdle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986-995. JSTOR: · Zbl 0703.62052 · doi:10.2307/2290074 · links.jstor.org
[7] Levey, A. S., Adler, S., Beck, G. J. et al. (1994). The effects of dietary protein restriction and blood pressure control on the progression of renal disease. N. Engl. J. Med. 330 877-884.
[8] Levey, A. S. et al. for the MDRD group (1999). A more accurate method to estimate glomerular filtration rate from serum creatinine: A new prediction equation. Ann. Intern. Med. 130 461-470.
[9] Ma, Y. C., Zuo, L., Chen, J. H. et al. (2006). Modified glomerular filtration rate estimating equation for chinese patients with chronic kidney disease. J. Am. Soc. Nephrol. 17 2937-2944.
[10] Owen, A. B. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90-120. · Zbl 0712.62040 · doi:10.1214/aos/1176347494
[11] Owen, A. (1991). Empirical likelihood for linear models. Ann. Statist. 19 1725-1747. · Zbl 0799.62048 · doi:10.1214/aos/1176348368
[12] Owen, A. B. (2001). Empirical Likelihood . Chapman & Hall, New York. · Zbl 0989.62019
[13] Prakasa Rao, B. L. S. (1983). Nonparametric Functional Estimation . Academic Press, New York. · Zbl 0542.62025
[14] Rosman, J. B., ter Wee, P. M., Meijer, S., Piers-Becht, T. P. M., Sluiter, W. J. and Donker, A. J. M. (1984). Prospective randomized trial of early protein restriction in chronic renal failure. Lancet 2 1291-1296.
[15] Sentürk, D. and Müller, H. G. (2005). Covariate-adjusted regression. Biometrika 92 75-89. · Zbl 1068.62082 · doi:10.1093/biomet/92.1.75
[16] Sentürk, D. and Müller, H. G. (2006). Inference for covariate-adjusted regression via varying coefficient models. Ann. Statist. 34 654-679. · Zbl 1095.62045 · doi:10.1214/009053606000000083
[17] Staniswalis, J. G. (2006). On fitting generalized nonlinear models with varying coefficients. Comput. Statist. Data Anal. 50 1818-1839. · Zbl 1445.62304
[18] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002
[19] Wu, J. F. (1981). Asymptotic theory of nonlinear least squares estimation. Ann. Statist. 9 501-513. · Zbl 0475.62050 · doi:10.1214/aos/1176345455
[20] Xue, L. G. and Zhu, L. X. (2007). Empirical likelihood for a varying coefficient model with longitudinal data. J. Amer. Statist. Assoc. 102 642-654. · Zbl 1172.62306 · doi:10.1198/016214507000000293 · caliban.asa.catchword.org
[21] Zhu, L. X. and Fang, K. T. (1996). Asymptotics for kernel estimate of sliced inverse regression. Ann. Statist. 24 1053-1068. · Zbl 0864.62027 · doi:10.1214/aos/1032526955
[22] Zhu, L. X. and Xue, L. G. (2006). Empirical likelihood confidence regions in a partially linear single-index model. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 549-570. · Zbl 1110.62055 · doi:10.1111/j.1467-9868.2006.00556.x
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.