Open Access
June, 1976 The Infinite Secretary Problem
Jacqueline Gianini, Stephen M. Samuels
Ann. Probab. 4(3): 418-432 (June, 1976). DOI: 10.1214/aop/1176996090


An infinite sequence of rankable individuals (rank $1 =$ best) arrive at times which are i.i.d., uniform on (0, 1). We, in effect, observe only their relative ranks as they arrive. We seek a stopping rule to minimize the mean of a prescribed positive increasing function, $q(\bullet)$, of the actual rank of the individual chosen. Let $f(t)$ be the minimal mean among all stopping rules which are greater than $t$. Then $f(\bullet)$ is a solution to a certain differential equation which is derived and used to find an optimal stopping rule. This problem is in a strong sense the "limit" of a corresponding sequence of "finite secretary problems" which have been examined by various authors. The limit of the "finite-problem" minimal risks is finite if and only if the differential equation has a solution, $f(\bullet)$, which is finite on $\lbrack 0, 1)$ with $f(1^-) = \sup q(n)$. Usually, if such a solution exists, it is unique, in which case $f(0)$ is both the minimal risk for the infinite problem and the limit of the "finite-problem" minimal risks.


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Jacqueline Gianini. Stephen M. Samuels. "The Infinite Secretary Problem." Ann. Probab. 4 (3) 418 - 432, June, 1976.


Published: June, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0341.60033
MathSciNet: MR413389
Digital Object Identifier: 10.1214/aop/1176996090

Primary: 60G40

Keywords: loss function , Optimal stopping rules , relative ranks

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • June, 1976
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