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.
Buonaguidi, B., Analysis of a sufficient statistic in a sequential detection problem, in Statistics for Innovation II (Italian Statistical Society Series on Advances in Statistics), (Genova, 16-18 June 2025), Springer Nature Switzerland AG, Gewerbestrasse 11, 6330 Cham, Switzerland 2025: 277-283. [10.1007/978-3-031-96303-2_45] [https://hdl.handle.net/10807/323280]
Analysis of a sufficient statistic in a sequential detection problem
Buonaguidi, Bruno
Primo
2025
Abstract
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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