## The Annals of Applied Probability

### Optimal selection problems based on exchangeable trials

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 6, Number 3 (1996), 862-882.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968230

Digital Object Identifier
doi:10.1214/aoap/1034968230

Mathematical Reviews number (MathSciNet)
MR1410118

Zentralblatt MATH identifier
0903.60034

#### Citation

Gnedin, Alexander V.; Krengel, Ulrich. Optimal selection problems based on exchangeable trials. Ann. Appl. Probab. 6 (1996), no. 3, 862--882. doi:10.1214/aoap/1034968230. https://projecteuclid.org/euclid.aoap/1034968230

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