Block Empirical Likelihood Inference for Longitudinal Generalized Partially Linear Single-Index Models

A key feature of longitudinal data is within-subject correlation, which, if neglected, can compromise efficiency and distort uncertainty quantification. Generalized estimating equations (GEE) (Liang and Zeger, 1986) offer a widely used semiparametric framework that avoids full likelihood specification by relying on moment restrictions and a working correlation. Subsequent developments clarified how to improve efficiency and robustness under correlation misspecification; for example, the quadratic inference function was proposed by Qu et al. (2000) to provide an alternative moment construction with favorable testing properties. When the mean structure departs from a purely parametric form, semiparametric regression for clustered outcomes using GEE becomes especially attractive (Lin and Carroll, 2001), and new estimation and model selection procedures for longitudinal semiparametric modeling were developed in Fan and Li (2004). Nevertheless, Wald-type inference based on sandwich covariance estimation can be unstable when the number of clusters is moderate and the working correlation is difficult to calibrate in practice (Liang, 2008).

Meanwhile, purely linear predictors can be too rigid for modern longitudinal studies, where covariate effects may be nonlinear, heterogeneous across subjects, or driven by a few latent directions. Single-index structures address this by projecting high-dimensional covariates onto a single informative index and estimating an unknown univariate link, with the choice of smoothing level playing a crucial role; Härdle et al. (1993) studied optimal smoothing for this class of models. Partially linear single-index models further retain an explicit linear component for interpretability while capturing remaining nonlinear variation through the index link, aligning naturally with additive and other nonparametric regression ideas (Stone, 1985). For independent data, Yu and Ruppert (2002) proposed penalized spline estimation procedures for partially linear single-index models, and Xia and Hardle (2006) developed semiparametric estimation theory that justifies their asymptotic properties. To accommodate non-Gaussian outcomes, Carroll et al. (1997) introduced the generalized partially linear single-index model (GPLSIM), which embeds the unknown link in a generalized mean structure and thus bridges generalized linear modeling with flexible regression. In repeatedmeasures settings, Liang et al. (2010) developed estimation and testing methods for partially linear single-index models with longitudinal data, while Bai et al. (2009) studied model-checking tools tailored to longitudinal single-index specifications. Closely related semiparametric longitudinal formulations include local polynomial mixed-effects models proposed by Wu and Zhang (2002) and polynomial spline inference for varying-coefficient models developed in Huang et al. (2004). Because longitudinal outcomes are often contaminated by outliers or heavy-tailed noise, robust alternatives have been pursued: Qin and Zhu (2008) investigated robust estimation in partial linear models with longitudinal data, and Liu and Lian (2018) studied robust procedures for varying-coefficient models in longitudinal settings.

Despite this extensive modeling literature, reliable inference for longitudinal GPLSIMs remains challenging because the unknown link function is a nuisance component whose estimation error can affect the second-order behavior of inference on finite-dimensional parameters. This difficulty is closely related to general principles for inference on parameters in semiparametric models (He and Shi, 2000), and becomes more pronounced when variable selection under correlation is also of interest. From an implementation standpoint, generalized semiparametric fitting is often carried out using iteratively reweighted least squares, as discussed by Green (1984), and stable quasi-Newton updating strategies can be helpful for high-dimensional optimization (Nocedal, 1980). Spline sieves provide a practical approximation device for unknown smooth functions De Boor (2001), and a comprehensive treatment of semiparametric regression is given by Ruppert et al. (2003). In modern longitudinal studies with dense trajectories, connections to principal component methodology for functional and longitudinal data also offer useful perspective on dimension reduction and variability (Hall et al., 2006).

These challenges motivate inferential approaches that remain faithful to the estimating-equation paradigm while avoiding unstable plug-in variance calculations. Empirical likelihood provides a convenient vehicle: it treats moment restrictions as the primitive object and yields likelihood-ratio type confidence regions without specifying a full parametric likelihood.