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## Discussiones Mathematicae Probability and Statistics

2014 | 34 | 1-2 | 5-21
Tytuł artykułu

### The "Thirty-seven Percent Rule" and the secretary problem with relative ranks

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We revisit the problem of selecting an item from n choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first k items and then select the first subsequent item that ranks lower than the l-th lowest-ranked item among the first k. We prove that the optimal rule has k ~ n/e, as in the classical secretary problem where our sole objective is to select the item of lowest rank; however, with the optimally chosen l, here we can get the expected rank of the item selected to be less than any positive power of n (as n approaches infinity). We also introduce a common generalization where our goal is to minimize the expected rank of the item selected, but this rank must be within the lowest d.
Słowa kluczowe
EN
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
5-21
Opis fizyczny
Daty
wydano
2014
otrzymano
2013-09-11
Twórcy
autor
• Department of Mathematics, Gettysburg College, 300 N. Washington Street, Gettysburg, PA 17325-1486 USA
autor
• Department of Economics, Boston University, 270 Bay State Rd, Boston, MA 02215 USA
Bibliografia
• [1] J. Bearden, A new secretary problem with rank-based selection and cardinal payoffs, J. Math. Psych. 50 (2006) 58-59. doi: 10.1016/j.jmp.2005.11.003
• [2] F. Bruss and T. Ferguson, Minimizing the expected rank with full information, J. Appl. Prob. 30 (1993) 616-626. doi: 10.2307/3214770
• [3] Y. Chow, S. Moriguti, H. Robbins and S. Samuels, Optimal selection based on relative ranks, Israel J. Math. 2 (1964) 81-90.
• [4] T. Ferguson, Who solved the secretary problem?, Statist. Sci. 4 (1989) 282-296. doi: 10.1214/ss/1177012493
• [5] P. Freeman, The secretary problem and its extensions - A review, Internat. Statist. Rev. 51 (1983) 189-206.
• [6] J. Gilbert and F. Mosteller, Recognizing the maximum of a sequence, J. Amer. Statist. Assoc. 61 (1966) 35-73. doi: 10.2307/2283044
• [7] A. Krieger and E. Samuel-Cahn, The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalizations, Adv. Appl. Prob. 41 (2009) 1041-1058. doi: 10.1239/aap/1261669585
• [8] D.V. Lindley, Dynamic programming and decision theory, Appl. Statistics 10 (1961) 39-51. doi: 10.2307/2985407
• [9] D. Pfeifer, Extremal processes, secretary problems and the 1/e law, J. Appl. Prob. 26 (1989) 722-733.
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Bibliografia
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