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Enhanced upper confidence limits via randomized tests in random sampling without replacement

Part of: Sampling theory, sample surveys Parametric inference

Published online by Cambridge University Press:  09 October 2025

Zihao Li Open the ORCID record for Zihao Li [Opens in a new window] ,
Huangjun Zhu Open the ORCID record for Huangjun Zhu [Opens in a new window]  and
Masahito Hayashi Open the ORCID record for Masahito Hayashi [Opens in a new window]
Zihao Li*
Affiliation:
Fudan University
Huangjun Zhu*
Affiliation:
Fudan University
Masahito Hayashi*
Affiliation:
The Chinese University of Hong Kong
*
*Postal address: Department of Physics and State Key Laboratory of Surface Physics, Institute for Nanoelectronic Devices and Quantum Computing, Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China.
*Postal address: Department of Physics and State Key Laboratory of Surface Physics, Institute for Nanoelectronic Devices and Quantum Computing, Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China.
****Postal address: School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen 518172, China; International Quantum Academy (SIQA), Futian District, Shenzhen 518048, China; and Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan. Email: hmasahito@cuhk.edu.cn
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Abstract

In this paper we study one-sided hypothesis testing under random sampling without replacement, which frequently appears in the cryptographic problem setting, including the verification of measurement-based quantum computation. Suppose that $n+1$ binary random variables $X_1,\ldots, X_{n+1}$ follow a permutation invariant distribution and n binary random variables $X_1,\ldots, X_{n}$ are observed. Then, we propose randomized tests with a randomization parameter for the expectation of the $(n+1)$th random variable $X_{n+1}$ under a given significance level $\delta>0$. Our randomized tests significantly improve the upper confidence limit over deterministic tests. Our problem setting commonly appears in machine learning in addition to cryptographic scenarios by considering adversarial examples. Such studies are essential for expanding the applicable area of statistics. Although this paper addresses only binary random variables, a similar significant improvement by randomized tests can be expected for general non-binary random variables.

Keywords

Randomized testrandom sampling without replacementadversarial scenarioadversarial example

MSC classification

Primary: 62F05: Asymptotic properties of tests 62D05: Sampling theory, sample surveys
Secondary: 62F03: Hypothesis testing

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Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 51
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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