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Optimal stopping rule for the full-information duration problem with random horizon

  • Mitsushi Tamaki (a1)


The full-information duration problem with a random number N of objects is considered. These objects appear sequentially and their values Xk are observed, where Xk, independent of N, are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the kth object is a relative maximum if Xk = max{X1, X2, . . ., Xk}). We assume that N is a random variable with a known upper bound n, so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is N or n. The structure of the optimal rule, which depends on the prior distribution assumed on N, is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that N / n converges, as n → ∞, to a random variable Vm having density fVm(v) = m(1 - v)m-1, 0 ≤ v ≤ 1 for a positive integer m. An interesting feature is that the optimal duration (relative to n) for Model 2 is just (m + 1) times as large as that for Model 1 asymptotically.


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* Postal address: Faculty of Business Administration, Aichi University, Nagoya Campus, Hiraike 4-60-6, Nakamura, Nagoya, Aichi, 453-8777, Japan. Email address:


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Ferguson, T. S., Hardwick, J. P. and Tamaki, M. (1992). Maximizing the duration of owning a relatively best object. In Strategies for Sequential Search and Selection in Real Time (Contemp. Math. 125), American Mathematical Society, Providence, RI, pp. 3757.
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 3573.
Gnedin, A. V. (1996). On the full information best-choice problem. J. Appl. Prob. 33, 678687.
Gnedin, A. V. (2004). Best choice from the planar Poisson process. Stoch. Process. Appl. 111, 317354.
Mazalov, V. V. and Tamaki, M. (2006). An explicit formula for the optimal gain in the full-information problem of owning a relatively best object. J. Appl. Prob. 43, 87101.
Porosinski, Z. (1987). The full-information best choice problem with a random number of observations. Stoch. Process. Appl. 24, 293307.
Samuel-Cahn, E. (1996). Optimal stopping with random horizon with application to the full-information best-choice problem with random freeze. J. Amer. Statist. Assoc. 91, 357364.
Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis, Dekker, New York, 381405.
Samuels, S. M. (2004). Why do these quite different best-choice problems have the same solutions? Adv. Appl. Prob. 36, 398416.
Tamaki, M. (2013). Optimal stopping rule for the no-information duration problem with random horizon. Adv. Appl. Prob. 45, 10281048.
Tamaki, M. (2015). On the optimal stopping problems with monotone thresholds. J. Appl. Prob. 52, 926940.


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Optimal stopping rule for the full-information duration problem with random horizon

  • Mitsushi Tamaki (a1)


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