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The full-information best choice problem with a random number of observations. (English) Zbl 0623.60059
Consider a number N of independent random variables having the same (known) distribution F; these random variables are observed sequentially, and one of them has to be selected. The objective is to maximize the probability of selecting the largest, and neither recall of past observations nor uncertainty concerning the availability of the selected observation are allowed. This problem was solved, first heuristically by J. P. Gilbert and F. Mosteller, J. Am. Stat. Assoc. 61, 35-73 (1966), and then rigorously by T. Bojdecki, Stochastic Processes Appl. 6, 153-163 (1978; Zbl 0374.60088), when N is a fixed integer.
The same problem is considered here, but N is now an integer-valued random variable. Like in the work by È. L. Presman and I. M. Sonin, Theory Probab. Appl. 17, 657-668 (1973); translation from Teor. Verojatn. Primen. 17, 695-706 (1972; Zbl 0296.60031), the optimal stopping rule is defined by a “stopping set” consisting of one or more “stopping islands”; in the single-island case, the optimal stopping rule is of the same form as that for the problem with N fixed. This single-island case is also a version of the “monotone case” of Y. S. Chow, H. Robbins and D. Siegmund, Great expectations: The theory of optimal stopping. (1971; Zbl 0233.60044).
As particular cases of the single-island case, the following distributions of N are considered:
i) N constant; ii) N uniform on \(\{\) 1,2,...,n\(\}\) ; iii) N Poisson with parameter \(\lambda\) ; iv) N negative binomial with parameters p and r; v) N has three possible values only; vi) N is geometric with parameter p.
In the geometric case, the optimal stopping time is explicitly derived, and in the uniform case, numerical results are given for \(n=21\) to 60.
Reviewer: J.Gianini-Pettitt

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
62L15 Optimal stopping in statistics
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References:
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