Odds algorithm: Difference between revisions

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==Applications==

{{secretary_problem.svg}}

Applications reach from medical questions in [[clinical trial]]s over sales problems, [[secretary problems]], [[portfolio (finance)|portfolio]] selection, (one way) search strategies, trajectory problems and the [[parking problem]] to problems in online maintenance and others.

There exists, in the same spirit, an Odds Theorem for continuous-time arrival processes with [[independent increments]] such as the [[Poisson process]] ({{harvnb|Bruss|2000}}). In some cases, the odds are not necessarily known in advance (as in Example 2 above) so that the application of the odds algorithm is not directly possible. In this case each step can use [[sequential estimate]]s of the odds. This is meaningful, if the number of unknown parameters is not large compared with the number n of observations. The question of optimality is then more complicated, however, and requires additional studies. Generalizations of the odds algorithm allow for different rewards for failing to stop

and wrong stops as well as replacing independence assumptions by weaker ones ({{harv|Ferguson (|2008))}}.

== Variations ==

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*{{cite journal |last1=Ano |first1=K.|first2=H. |last2=Kakinuma |first3=N. |last3=Miyoshi |title=Odds theorem with multiple selection chances |journal=Journal of Applied Probability |volume=47 |year=2010 |issue=4|pages=1093–1104 |doi=10.1239/jap/1294170522|s2cid=17598431|doi-access=free }}

* {{cite journal | last=Bruss | first=F. Thomas | authorlink=Franz Thomas Bruss | title=Sum the odds to one and stop | journal=The Annals of Probability | publisher=Institute of Mathematical Statistics | volume=28 | issue=3 | year=2000 | issn=0091-1798 | doi=10.1214/aop/1019160340 | pages=1384–1391 | doi-access=free }}

* —:* "[https://www.cambridge.org/core/journals/journal-of-applied-probability/article/note-on-a-lower-bound-for-the-multiplicative-odds-theorem-of-optimal-stopping/B759B6E4A9D83DB84D6EE1B4C827785B A note on Bounds for the Odds Theorem of Optimal Stopping]", ''[[Annals of Probability]]'' Vol. 31, 1859–1862, (2003).

* —:* "[https://web.archive.org/web/20230409100437/https://www.ems-ph.org/journals/newsletter/pdf/2006-12-62.pdf The art of a right decision]", ''Newsletter of the [[European Mathematical Society]]'', Issue 62, 14–20, (2005).

*{{wikicite |ref={{SfnRef|Ferguson|2008}} |reference=[[Thomas S. Ferguson|T. S. Ferguson]]: (2008, unpublished)}}{{sfn whitelist|CITEREFFerguson2008}}

*{{cite journal |first1=F. T. |last1=Bruss |first2=D. |last2=Paindaveine |title=Selecting a sequence of last successes in independent trials |journal=Journal of Applied Probability |volume=37 |pages=389–399 |year=2000 |issue=2 |doi=10.1239/jap/1014842544 |url=https://mpra.ub.uni-muenchen.de/21166/1/MPRA_paper_21166.pdf}}

*{{Cite journal|last1=Gilbert |first1=J |last2=Mosteller |first2=F |title= Recognizing the Maximum of a Sequence |journal=Journal of the American Statistical Association |volume=61 |pages=35–73 |year=1966 |issue=313 |doi=10.2307/2283044 |jstor=2283044 }}