The full-information best choice problem with a random number of observations
type: Item
instance of: scholarly article
title: The full-information best choice problem with a random number of observations
zbMATH Open document ID: 0623.60059
DOI: 10.1016/0304-4149(87)90020-2
author: Zdzisław Porosiński
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: 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 \textit{J. P. Gilbert} and \textit{F. Mosteller}, J. Am. Stat. Assoc. 61, 35-73 (1966), and then rigorously by \textit{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 \textit{È. L. Presman} and \textit{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 \textit{Y. S. Chow}, \textit{H. Robbins} and \textit{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.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 62L15
zbMATH DE Number: 4011569
zbMATH Keywords: best choice problem
zbMATH Keywords: secretary problem
zbMATH Keywords: optimal stopping rule
zbMATH Keywords: numerical results
reviewed by: J. Gianini Pettitt
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90020-2
OpenAlex ID: W2076888737
cites work: On optimal stopping of a sequence of independent random variables- probability maximizing approach
cites work: Q3896307
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cites work: The Secretary Problem and Its Extensions: A Review
cites work: The Best Choice Problem for a Random Number of Objects
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cites work: Investigations by statistical sequential analysis
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description: scientific article; zbMATH DE number 4011569