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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 62G20 Nonparametric asymptotic efficiency 62J02 General nonlinear regression 65C60 Computational problems in statistics