Unified inference for longitudinal/functional data quantile dynamic additive models

We investigate the unified inference of a time-varying additive model under the quantile regression framework, considering both sparse and dense longitudinal or functional data. For convolution-type smoothed objective functions, we propose a two-step method for estimating both the trend and the component functions. Theoretical analysis shows that the two-step estimators share the same asymptotic distribution as the oracle estimators, while the convergence rates and limiting variance functions differ between sparse and dense situations. However, making a subjective choice between these two cases can lead to incorrect statistical inferences. To address this issue, we develop sandwich formulas for variance estimations. This allows us to establish a unified inference without the need to decide whether the data are sparse or dense. Via simulation studies, we assess the finite-sample performance of the proposed methods. Finally, analyses of two different types of real data illustrate our proposed methods.