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The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalizations

Part of: Sequential methods Limit theorems Parametric inference

Published online by Cambridge University Press:  01 July 2016

Abba M. Krieger  and
Ester Samuel-Cahn
Abba M. Krieger*
Affiliation:
University of Pennsylvania
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address: Department of Statistics, The Wharton School University of Pennsylvania, Philadelphia, PA 19104-6340, USA. Email address: krieger@wharton.upenn.edu
∗∗ Postal address: Department of Statistics (and Center for Rationality), The Hebrew University of Jerusalem, Jerusalem, 91905, Israel. Email address: scahn@mscc.huji.ac.il
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Abstract

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The secretary problem for selecting one item so as to minimize its expected rank, based on observing the relative ranks only, is revisited. A simple suboptimal rule, which performs almost as well as the optimal rule, is given. The rule stops with the smallest i such that Ri≤ic/(n+1-i) for a given constant c, where Ri is the relative rank of the ith observation and n is the total number of items. This rule has added flexibility. A curtailed version thereof can be used to select an item with a given probability P, P<1. The rule can be used to select two or more items. The problem of selecting a fixed percentage, α, 0<α<1, of n, is also treated. Numerical results are included to illustrate the findings.

Keywords

Secretary problemgroup selection rulessuboptimal rulenear optimal ruleabsolute rankrelative rank60F15

MSC classification

Primary: 62L99: None of the above, but in this section
Secondary: 62F07: Ranking and selection 60F15: Strong theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1041 - 1058
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported by funds from the Marcy Bogen Chair of Statistics at the Hebrew University of Jerusalem.

Supported by the Israel Science Foundation Grant 467/04.

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