Empirical likelihood inference in partially linear single-index models for longitudinal data

• Published in: Journal of multivariate analysis Vol. 101; no. 3; pp. 718 - 732
• Main Authors: Li, Gaorong, Zhu, Lixing, Xue, Liugen, Feng, Sanying
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.03.2010; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Algorithms; Approximation; Bias; Bias correction; Confidence intervals; Confidence region; Correlation analysis; Empirical likelihood; Exact sciences and technology; Linear inference, regression; Longitudinal data; Longitudinal data Partially linear single-index model Empirical likelihood Confidence region Bias correction; Mathematics; Monte Carlo simulation; Multivariate analysis; Nonparametric inference; Parameter estimation; Parametric inference; Partially linear single-index model; Probability and statistics; Sciences and techniques of general use; Statistics; Studies; secondary; Bias correction; Confidence region; Empirical likelihood; Longitudinal data; Partially linear single-index model; primary; Biometrics; Normal approximation; Correlation; Parameter estimation; Statistical distribution; Bias; primary 62J05; Non parametric estimation; Iterative method; Multivariate analysis; Stochastic method; Statistical simulation; Epidemiology; Linear model; Medical science; Consistent estimator; Approximation theory; Method study; Likelihood function; Monte Carlo method; Asymptotic behavior; Empirical method; Statistical association; Statistical estimation; Covariance matrix; Algorithm; Chi square; Confidence interval; Statistical method; Numerical analysis; Correlation analysis; secondary 62G20; Biased estimation; 62G15
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2009.08.006

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.