Bandit Process Inference for Streaming Functional Data

The functional linear model Y=∫TX(t)β0(t)dt+ε is widely used in various fields, including medical image analysis, economics, and climate science. Reliable statistical inference for β0(t) across the domain t∈T is essential in these applications. While substantial progress has been made in addressing prediction problems in offline settings, hypothesis testing for β0(t), especially in the context of streaming and functional data, continues to present considerable challenges. In the paper, we propose a novel statistical inference framework designed specifically for streaming data based on functional stochastic gradient descent algorithms. Our approach constructs an online sequence of randomly perturbed estimators and incorporates a two-armed bandit process to build an adaptive test statistic for hypothesis testing. Moreover, we establish the limiting distribution of this statistic under both the null and alternative hypotheses, thus ensuring its statistical validity. Simulation studies examine the power and null calibration of the framework, and a real-world application illustrates its use for streaming functional data.