Unified inference in generalized varying coefficient longitudinal data model with unknown link function

In this paper, we investigate the estimation and unified inference for varying coefficient models with an unspecified link function, accommodating both sparse and dense longitudinal data structures. Utilizing the local smoothing techniques, we propose an iterative algorithm to estimate the varying coefficients and the link function. We establish the asymptotic normality of the resulting local linear estimators under both sparse and dense sampling schemes, revealing distinct theoretical behaviours across these settings. Given that the distinction between sparse and dense designs is often ambiguous in applications, we further derive a unified asymptotic distribution for the estimated nonparametric coefficient functions and link function. This enables the development of a robust inference framework that remains valid across a spectrum of data densities. Finally, the finite-sample performance of the proposed methodology is demonstrated through Monte Carlo simulations and empirical data analysis.