Abstract
We consider optimal stopping problems with loss function q depending on the rank of the stopped random variable. Samuels asked whether there exists an exchangeable sequence of random variables $X_1, \dots, X_n$ without ties for which the observation of the values of the $X_i$'s gives no advantage in comparison with the observation of just the relative ranks of the variables. We call distributions of the sequences with this property q-noninformative and derive necessary and sufficient conditions for this property. Extending an impossibility result of B. Hill, we show that, for any $n > 1$, there are certain losses q for which q-noninformative distributions do not exist. Special attention is given to the classical problem of minimizing the expected rank: for n even we construct explicitly universal randomized stopping rules which are strictly better than the rank rules for any exchangeable sequence.
Citation
Alexander V. Gnedin. Ulrich Krengel. "Optimal selection problems based on exchangeable trials." Ann. Appl. Probab. 6 (3) 862 - 882, August 1996. https://doi.org/10.1214/aoap/1034968230
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