Estimation and Inference for One-pass Streaming Function-on-Function Regression

In many applications, functional pairs are collected sequentially from a data stream. Existing function-on-function regression methods are primarily designed for batch data, and bootstrap inference has become one of the most popular approaches. However, this procedure may be infeasible when observations cannot be stored. In this paper, we propose a one-pass functional SGD estimator that updates upon each observation's arrival without data storage, and a weighted rolling validation rule is provided for step size selection for implementation. Further, an online multiplier bootstrap is developed for inference that updates perturbed stochastic gradient paths in parallel, bypassing estimating an infinite-dimensional covariance operator. Asymptotic behaviors are established for the proposed method, and conditional bootstrap validity is justified for the bias-centered pointwise estimation error. Simulation studies and two real data applications confirm the empirical effectiveness of the proposed procedures. To the best of our knowledge, this is the first theoretically justified one-pass framework for bootstrap inference on a bivariate coefficient surface with streaming functional data.