Online robust estimation and bootstrap inference for function-on-scalar regression

TL;DR

This paper introduces a robust online function-on-scalar regression method using geometric median and stochastic gradient descent, enabling efficient analysis of streaming data with reliable inference.

Contribution

It presents a novel online estimation technique with theoretical guarantees and a bootstrap-based inference procedure for functional data analysis.

Findings

Method is computationally efficient and scalable.

Theoretical properties like consistency and asymptotic normality are established.

Numerical and real data studies demonstrate effectiveness and practical utility.

Abstract

We propose a novel and robust online function-on-scalar regression technique via geometric median to learn associations between functional responses and scalar covariates based on massive or streaming datasets. The online estimation procedure, developed using the average stochastic gradient descent algorithm, offers an efficient and cost-effective method for analyzing sequentially augmented datasets, eliminating the need to store large volumes of data in memory. We establish the almost sure consistency, $L_p$ convergence, and asymptotic normality of the online estimator. To enable efficient and fast inference of the parameters of interest, including the derivation of confidence intervals, we also develop an innovative two-step online bootstrap procedure to approximate the limiting error distribution of the robust online estimator. Numerical studies under a variety of scenarios demonstrate the effectiveness and efficiency of the proposed online learning method. A real application analyzing PM$_{2.5}$ air-quality data is also included to exemplify the proposed online approach.