The Recent History Functional Linear Model and Its Extension to Sparse Longitudinal Data

We propose a variant of historical functional linear models for cases where the current response is affected by the predictor process in a window into the past. Different from the rectangular support of functional linear models, the triangular support of the historical functional linear models and the point-wise support of varying coefficient models, the current model has a sliding window support into the past. This idea leads to models that bridge the gap between the varying coefficient models and the functional linear (historic) models for densely measured functional data and longitudinal data. By utilizing one dimensional basis expansions and one dimensional smoothing procedures, the proposed estimation algorithm is shown to have better performance and to be faster than the estimation procedures proposed for historical functional linear models. We also consider the recent history functional linear models, relating a longitudinal response to a longitudinal predictor. We propose an estimation procedure for recent history functional linear models that is geared towards sparse longitudinal data, where the observation times across subjects are irregular and total number of measurements per subject is small. The proposed estimation procedure builds upon recent developments in literature for estimation of functional linear models with sparse data and utilizes connections between the recent history functional linear models and the varying coefficient models. We establish uniform consistency of the proposed estimators, propose prediction of the response trajectories and derive their asymptotic distribution leading to asymptotic point-wise confidence bands. We include a real data application and simulation studies to demonstrate the efficacy of the proposed methodology.