Abstract

In this study, we introduce an inferential procedure for assessing the covariance

difference between two-samples of large-scale functional data, utilizing a computationally efficient multiplier bootstrap approach. In contrast to the existing method that focuses

exclusively on a testing procedure, our approach starts by establishing a confidence region

for the covariance difference under fairly flexible conditions. This leads not only to a more

powerful test but also to an accessible estimated power function, which is shown consistent

across a broad range of alternatives. A notable characteristic of the new procedure is that it

requires less stringent distributional assumptions for large-scale functional data, and does

not impose any structural constraints on covariances or correlations. Moreover, the proposed procedure benefits from the desirable properties of being “eigenvalue-decay-free”

and “square-integrable-free”, tailored for functional data. We conduct a simulation study

and a real data application to assess the numerical performance of the proposed method.

Key words and phrases: Functional data; Ultra-high-dimensionality; Covariance function; Confidence region; Multiplier bootstrap

Information

Preprint No.SS-2025-0295
Manuscript IDSS-2025-0295
Complete AuthorsKaijie Xue, Lan Xue, Riquan Zhang
Corresponding AuthorsRiquan Zhang
Emailszhangriquan@163.com

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Acknowledgments

Kaijie Xue and Lan Xue are the co-first authors. Riquan Zhang is the corical order, and all authors have made equal contributions. This research was

supported by the National Natural Science Foundation of China (12371268,

12531013, 12371272), the Youth Project of Shanghai Eastern Talent Program

(QNJY2024152), the Basic Research Project of Shanghai Science and Technology Commission (22JC1400800), and awards from the National Institutes of

Diabetes, Digestive, and Kidney Disease Award numbers R01DK132385.

Supplementary Materials

The auxiliary lemmas with their proofs, along with the proofs of the main

theorems, are relegated to an online Supplementary Material to save space.

Supplementary materials are available for download.