The Annals of Statistics

A statistical version of prophet inequalities

David Assaf, Larry Goldstein, and Ester Samuel-Cahn

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All classical “prophet inequalities” for independent random variables hold also in the case where only a noise-corrupted version of those variables is observable. That is, if the pairs $(X_1, Z_1),\ldots,(X_n, Z_n)$ are independent with arbitrary, known joint distributions, and only the sequence $Z_1 ,\ldots,Z_n$ is observable, then all prophet inequalities which would 1 n hold if the $X$’s were directly observable still hold, even though the expected $X$-values (i.e., the payoffs) for both the prophet and statistician, will be different. Our model includes, for example, the case when $Z_i=X_i + Y_i$, where the $Y$’s are any sequence of independent random variables.

Article information

Ann. Statist., Volume 26, Number 3 (1998), 1190-1197.

First available in Project Euclid: 21 June 2002

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Zentralblatt MATH identifier

Primary: 62L15: Optimal stopping [See also 60G40, 91A60] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Prophet inequalities noisy observations perfect prophet optimal stopping


Assaf, David; Goldstein, Larry; Samuel-Cahn, Ester. A statistical version of prophet inequalities. Ann. Statist. 26 (1998), no. 3, 1190--1197. doi:10.1214/aos/1024691094.

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