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On optimal choosing of one of the k best objects

Author

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  • Porosinski, Zdzislaw

Abstract

A full-information continuous-time best choice problem is considered. A stream of objects being iid random variables with a known continuous distribution function is observed. The objects appear according to some renewal process independent of objects. The objective is to maximize the probability of selecting of one of the k best objects when observation is perfect, one choice can be made and neither recall nor uncertainty of selection is allowed. The horizon of observation is a positive random variable independent of objects. The natural case of a Poisson renewal process (with intensity [lambda]) and of exponentially distributed horizon (with parameter [mu]) is examined in detail. An optimal stopping rule stops at the first object which is greater than some constant level c(p) depending only on p=[mu]/([mu]+[lambda]). The probability of choosing the proper object P(win) is constant for all natural cases, i.e. when p is small. Simple formulae and numerical values for c(p) and P(win) are obtained. It is interesting that if p tends to 0, P(win) goes to 1 and c(p) goes to 0 at a much slower rate than exponentially fast.

Suggested Citation

  • Porosinski, Zdzislaw, 2003. "On optimal choosing of one of the k best objects," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 419-432, December.
    • Handle: RePEc:eee:stapro:v:65:y:2003:i:4:p:419-432
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    References listed on IDEAS

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    1. Bojdecki, Tomasz, 1978. "On optimal stopping of a sequence of independent random variables -- probability maximizing approach," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 153-163, January.
    2. Frank, Arthur Q. & Samuels, Stephen M., 1980. "On an optimal stopping problem of Gusein-Zade," Stochastic Processes and their Applications, Elsevier, vol. 10(3), pages 299-311, October.
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