On the asymptotic properties of the shrinkage empirical likelihood estimators for longitudinal data

When there exist time-dependent covariates in some longitudinal study, it is well-known that the widely used generalized estimating equations approach would not preserve unbiasedness and robustness in an arbitrary working correlation structure. However, incorrect application of the working correlation structure could result in loss of efficiency and biased estimation. To deal with this problem, Leung et al. proposed a shrinkage empirical likelihood approach which combines the unbiased estimating equations and the extracted additional information from the estimating equations that excluded by the independence assumption. Although their simulations have shown the proposed estimators are efficient, the asymptotic properties of the proposed estimators are unknown. Here it is was shown that the proposed estimators are consistent and asymptotically normally distributed under some regular conditions.