  Estimation and inference with modern longitudinal data from wearable devices,
which consist of biological signals at high-frequency time points, is burdened
by massive computational costs. We propose a distributed estimation and
inference procedure that efficiently estimates both functional and scalar
parameters with intensively measured longitudinal outcomes. The procedure
overcomes computational difficulties through a scalable divide-and-conquer
algorithm that partitions the outcomes into smaller sets. We circumvent
traditional basis selection problems by analyzing data using quadratic
inference functions in smaller subsets such that the basis functions have a low
dimension. To address the challenges of combining estimates from dependent
subsets, we propose a statistically efficient one-step estimator derived from a
constrained generalized method of moments objective function with a smoothing
penalty. We show theoretically and numerically that the proposed estimator is
as statistically efficient as non-distributed alternative approaches and more
efficient computationally. We demonstrate the practicality of our approach with
the analysis of accelerometer data from the National Health and Nutrition
Examination Survey.
