The full-information best choice problem with a random number of observations.

*(English)*Zbl 0623.60059Consider 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.

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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\textit{Z. Porosiński}, Stochastic Processes Appl. 24, 293--307 (1987; Zbl 0623.60059)

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##### References:

[1] | Bojdecki, T., On optimal stopping of a sequence of independent random variables—probability maximizing approach, Stochastic processes appl., 6, 153-163, (1978) · Zbl 0374.60088 |

[2] | Bojdecki, T., On optimal stopping of independent random variables appearing to a renewal process with random time horizon, Bol. soc. mat. Méxicana, 22, 35-40, (1977) · Zbl 0449.60032 |

[3] | Cowan, R.; Zabczyk, J., An optimal selection problem associated with the Poisson process, Theory prob. appl., 23, 606-614, (1978) · Zbl 0396.62063 |

[4] | Chow, Y.S.; Robbins, H.; Siegmund, D., Great expectations: the theory of optimal stopping, (1971), Houghton Mifflin Co Boston · Zbl 0233.60044 |

[5] | Freeman, P.R., The secretary problem and its extensions: A review, Int. stat. rev., 51, 189-206, (1983) · Zbl 0516.62081 |

[6] | Gilbert, J.; Mosteller, F., Recognizing the maximum of a sequence, J. amer. stat. assoc., 61, 35-73, (1966) |

[7] | Presman, E.L.; Sonin, I.M., The best choice problem for a random number of objects, Theory prob. appl., 17, 657-668, (1972) · Zbl 0296.60031 |

[8] | Sakaguchi, M., Optimal stopping problems for randomly arriving offers, Math. japonicae, 21, 201-217, (1976) · Zbl 0353.93054 |

[9] | Shiryaev, A.N., Statistical sequential analysis, (1969), Nauka Moscow, (in Russian) · Zbl 0169.21202 |

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