Odds Theorem with Multiple Selection Chances

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and fin...

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Bibliographic Details
Published in:Journal of applied probability 2010-12, Vol.47 (4), p.1093-1104
Main Authors: Ano, Katsunori, Kakinuma, Hideo, Miyoshi, Naoto
Format: Article
Language:English
Subjects:
60G40
62L15
Applications
Brasses
Computation
Conditional probabilities
Exact sciences and technology
Exact solutions
Failure
Induction assumption
Mathematical analysis
Mathematical theorems
Mathematics
multiple selection chances
Optimal stopping
Optimization
Parametric inference
Probabilities
Probability and statistics
Probability distribution
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
selecting the last success
Sequential methods
Statistics
Stochastic processes
Studies
Success
Theorems
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Summary:We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1294170522