Simultaneous Inference for Partially Observed Functional Time Series

2026-06-30

AI Summary

This study addresses the challenge of simultaneous inference for intermittently missing dependent functional time series, a setting where existing methods often fail due to their neglect of path dependence and the missingness mechanism. Departing from conventional Hilbert space frameworks, this work develops a novel approach within the space of bounded functions equipped with the supremum norm. By integrating Gaussian approximation and stochastic process theory, it achieves, for the first time, uniform simultaneous inference over the entire domain for partially observed functional time series—yielding simultaneous confidence bands without relying on the functional central limit theorem. The method is further enhanced with multiscale inference techniques to effectively detect nonstationary trends. Applied to urban air pollution data, it accurately identifies periods exceeding regulatory thresholds, offering the first practical tool for simultaneous inference in partially observed functional time series.

Abstract

Functional data analysis (FDA) provides statistical methods for analyzing samples of time-continuous stochastic processes. Measurements often arise in the form of sensor data for a key scientific variable. The practical problem of irregular sensor disruptions has fostered interest in analyzing partially observed random functions. Specifically, this paper is motivated by a time series of intermittently missing pollution data with dependence along pollution paths and missingness patterns. To allow statistical analysis, we develop the first inference methods for dependent, partially observed functional time series. Existing methods were not appropriate for this task, because they heavily rely on the independence of the data functions. Mathematically, we model data on the space of bounded functions equipped with the supremum norm. This allows simultaneous inference across the entire functional domain, including simultaneous confidence bands -- something existing Hilbert-space-based methods cannot provide. To study non-stationary trends along the time series, we extend state-of-the-art multiscale inference methods (originally developed for scalar data) to partially observed functions. The key application of the latter methods is testing for excessive pollution levels in inner cities. Our approach combines state-of-the-art Gaussian approximations with stochastic process theory. Interestingly, it also improves existing results for fully observed functional time series by avoiding a functional CLT.

Problem

• partially observed functional data
• functional time series
• statistical inference
• irregular missingness
• dependence

Innovation

• partially observed functional time series
• simultaneous inference
• supremum norm
• multiscale inference
• Gaussian approximation