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Optimal selection problems based on exchangeable trials. (English) Zbl 0903.60034

Some considerations related to the secretary problem [cf. T. S. Ferguson, Stat. Sci. 4, No. 3, 282-296 (1989; Zbl 0788.90080)] are given. It is the class of optimal stopping problems with loss function \(q\) dependent on the rank of the stopped random variable. S. M. Samuels [“Sufficiently noninformative priors for the secretary problem; the case \(n=3\)” (Preprint, Dept. Statistics, Pardue Univ., 1994)] has called distributions of an exchangeable sequence of random variables \(X_1,\ldots,X_n\) 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, \(q\)-noninformative. The following questions have been posed: (1) For a given loss function \(q\) does there exist a \(q\)-noninformative distribution for the \(\{X_i\}\) such that the minimal risk using stopping rules based on the relative ranks only is the same as that for the wider class of stopping rules adapted to the natural filtration of the sequence? (2) Is there a stopping rule, may be randomized, with respect to the larger filtration which performs better than any rule just based on the relative ranks for any distribution of the sequence?
Necessary and sufficient conditions for overall optimality of a rank rule in terms of some inequalities on predictive probabilities of relative ranks are given. It is shown that for any \(n\), there exist losses \(q\), for which the first question has a negative answer. Special attention is given to the classical problem of minimizing the expected rank: for \(n\) even, explicitly universal (randomized) stopping rules which are strictly better than the rank rules for any exchangeable sequence, have been constructed.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory

Citations:

Zbl 0788.90080
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References:

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