# Online Estimation for Functional Data

arXiv: https://arxiv.org/abs/2111.02750 · HTML: https://mlsanity.com/paper/2111.02750/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2021.

## Built on by (most cited first, 4 of 4)

- [Online Estimation with Rolling Validation: Adaptive Nonparametric Estimation with Streaming Data](https://mlsanity.com/paper/2310.12140/) (2023)
- [Adaptive debiased SGD in high-dimensional GLMs with streaming data](https://mlsanity.com/paper/2405.18284/) (2024)
- [Online Learning of Functional Principal Component Analysis for Multidimensional Functional Data](https://mlsanity.com/paper/2505.02131/) (2025)
- [Statistical Theory of Multi-stage Newton Iteration Algorithm for Online Continual Learning](https://mlsanity.com/paper/2508.07419/) (2025)

## Abstract (arXiv)

Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings.