Unified inference for longitudinal/functional data quantile dynamic additive models

Abstract 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.