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Empirical likelihood inference in partially linear single-index models for longitudinal data. (English) Zbl 1181.62034
Summary: The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.

##### MSC:
 62G05 Nonparametric estimation 62J05 Linear regression; mixed models 62G15 Nonparametric tolerance and confidence regions 62G20 Asymptotic properties of nonparametric inference 62G08 Nonparametric regression and quantile regression 65C60 Computational problems in statistics (MSC2010)
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