Large Language Models (LLMs) are rapidly gaining enormous popularity in recent years. However, the training of LLMs has raised significant privacy and legal concerns, particularly regarding the inclusion of copyrighted materials in their training data without proper attribution or licensing, which falls under the broader issue of data misappropriation. In this article, we focus on a specific problem of data misappropriation detection, namely, to determine whether a given LLM has incorporated data generated by another LLM. To address this issue, we propose embedding watermarks into the copyrighted training data and formulating the detection of data misappropriation as a hypothesis testing problem. We develop a general statistical testing framework, construct a pivotal statistic, determine the optimal rejection threshold, and explicitly control the type I and type II errors. Furthermore, we establish the asymptotic optimality properties of the proposed tests, and demonstrate its empirical effectiveness through intensive numerical experiments.
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