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Statistical inference for mean function of partially observed functional time series.

Shuang Sun1, Leheng Cai2, Qirui Hu3,4

  • 1Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, 19104, Philadelphia, USA.

Biometrics
|June 22, 2026
PubMed
Summary

We present a statistical framework for analyzing partially observed functional time series data. Our methods provide reliable statistical inference, including confidence bands and hypothesis tests, even with noisy measurements.

Keywords:
Skorokhod spacefunctional datamissing datarelevant hypothesis testsimultaneous confidence band

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Area of Science:

  • Statistics
  • Time Series Analysis
  • Functional Data Analysis

Background:

  • Functional time series data are increasingly common in various scientific fields.
  • Analyzing partially observed or noisy functional data presents significant statistical challenges.
  • Existing methods often struggle with the complexities of incomplete or error-prone functional time series.

Purpose of the Study:

  • To develop a robust statistical framework for inference on mean functions of partially observed functional time series.
  • To address both ideal (noiseless) and practical (noisy) scenarios of data observation.
  • To enable advanced statistical inference techniques tailored for functional time series.

Main Methods:

  • Establish weak convergence for an ideal estimator in the Skorokhod space for noiseless data.
  • Propose a B-spline estimator for noisy data and derive its asymptotic distribution using Gaussian approximation.
  • Utilize multiplier bootstrap for approximating limiting distributions and ensuring procedure consistency.

Main Results:

  • Theoretical results establish the asymptotic behavior of the proposed estimators.
  • Development of simultaneous confidence bands, two-sample tests, and hypothesis testing under the supremum norm.
  • Demonstration of the multiplier bootstrap's consistency for practical implementation.

Conclusions:

  • The developed statistical framework provides a powerful tool for analyzing partially observed functional time series.
  • The proposed methods are validated through numerical experiments and applied successfully to real-world electroencephalogram data.
  • The study contributes novel statistical inference techniques applicable to complex functional data analysis.