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In the problem of detecting the moment at which a Brownian motion, observed in real time, changes its drift, the optimal estimator of this moment is the stopping time given by the first instant at which a certain functional of the Brownian motion without drift exceeds a precomputed threshold. Then, as the decision on when to stop the observation of the Brownian motion depends only on this functional, the latter represents a sufficient statistic for the problem. For this sufficient statistic, we revise its basic properties, derive its density function, and discuss a method for the numerical evaluation of this density.

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Analysis of a Sufficient Statistic in a Sequential Detection Problem

  • 会议论文

  • 发表日期：

    2025年

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  • Bruno Buonaguidi

摘要

In the problem of detecting the moment at which a Brownian motion, observed in real time, changes its drift, the optimal estimator of this moment is the stopping time given by the first instant at which a certain functional of the Brownian motion without drift exceeds a precomputed threshold. Then, as the decision on when to stop the observation of the Brownian motion depends only on this functional, the latter represents a sufficient statistic for the problem. For this sufficient statistic, we revise its basic properties, derive its density function, and discuss a method for the numerical evaluation of this density.