Abstract

This paper considers the problem of testing the marginal distributions

of multiple, independent data streams, where for each data stream, multiple composite hypotheses along with an indifference zone are posed. A novel global error

metric is proposed, which aims to control the probabilities of making different

numbers of misclassifications below different, user-specified levels, and which includes the classical and the generalized misclassification probabilities as special

cases. A novel testing procedure is designed and is shown to achieve the minimum expected sample size under all possible distributions, among all tests that

control this global error metric below the same levels, asymptotically as any of

these levels goes to zero. This asymptotic optimality theory is established allowing temporal dependence and general information functions beyond linear that

are considered in most literature. Examples are provided to illustrate the theory

and numerical studies are presented to visualize both the asymptotic properties

and finite-sample performance.

*Author's ORCID ID: 0000 0001 8508 7982

Key words and phrases: asymptotic optimality, multihypothesis testing, multiple testing, non-linear information function, sequential analysis

Information

Preprint No.SS-2025-0042
Manuscript IDSS-2025-0042
Complete AuthorsYiming Xing
Corresponding AuthorsYiming Xing
Emailsyimingx4@tongji.edu.cn

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Acknowledgments

The author would like to thank the Editor, the Associate Editor and the

reviewers for valuable comments and constructive suggestions, which have

greatly improved the paper.

The author is supported by National Natural Science Foundation of China (No. 12501379), Shanghai Rising-Star

Program (No. 24YF2748500), and Open Research Fund of Key Laboratory

of Advanced Theory and Application in Statistics and Data Science (East

China Normal University), Ministry of Education (No. KLATASDS2501).

Supplementary Materials

In the supplementary material, we illustrate the general theory through

three concrete examples, present extra numerical studies of testing the correlation coefficient of autoregressive data, and present all proofs and a dis-

cussion about model misspecification.

Supplementary materials are available for download.