Geodesic
Duration problem: basic concept and some extensions
Mathematica Applicanda, Tome 44 (2016) no. 1, pp. 87-112.

Voir la notice de l'article provenant de la source Annales Societatis Mathematicae Polonae Series

We consider a sequence of independent random variables with the known distribution observed sequentially. The observation n is a value of one order statistics s : n-th, where 1 ≤ s ≤ n. It the instances following the n-th observation it may remain of the s : m or it will be the value of the order statistics r : m (of m > n observations). Changing the rank of the observation, along with expanding a set of observations is a random phenomenon that is difficult to predict. From practical reasons it is of great interest. Among others, we pose the question of the moment in which the observation appears and whose rank will not change significantly until the end of sampling of a certain size. We also attempt to answer which observation should be kept to have the "good quality observation" as long as possible. This last question was analysed by Ferguson, Hardwick and Tamaki (1991) in the abstract form which they called the problem of duration.This article gives a systematical presentation of known duration models and some new generalization. We collect results from different papers on the duration of the extremal observation in the no-information (say rank based) case and the full-information case. In the case of non-extremal observation duration models the most appealing are various setting related to the two extremal order statistic. In the no-information case it will be the maximizing duration of owning the relatively the best or the second best object. The idea was formulated and the problem was solved by Szajowski and Tamaki (2006). The full-information duration problem with special requirement was presented by Kurushima and Ano (2010).
DOI : 10.14708/ma.v44i1.829
Classification : 60G40, 62L15
Mots-clés : optimal stopping, duration problem, secretary problem 
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Zdzisław Porosiński; Marek Skarupski; Krzysztof Szajowski. Duration problem: basic concept and some extensions. Mathematica Applicanda, Tome 44 (2016) no. 1, pp.  87-112. doi : 10.14708/ma.v44i1.829. http://geodesic.mathdoc.fr/articles/10.14708/ma.v44i1.829/

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