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Remarks on the secretary problem. (English) Zbl 0749.62053
The classical secretary problem is considered: A given number of candidates are to be interviewed for a certain position. They present themselves one by one in random order. We can either hire the $$i$$-th candidate according to the (relative) rank among those candidates already interviewed, in which case the process stops, or go on to the next, in which case the $$i$$-th cannot be recalled. One of the candidates has to be selected. If the (absolute) rank of the selected one among all candidates is $$x$$, our loss will be $$f(x)$$, where $$f$$ is a real function defined on the positive integers. We want to choose a stopping rule that will minimize the expected loss (asymptotically as $$n\to\infty$$).
The following loss functions are dealt with: $$f(x) = 0$$ if $$x=1$$ and $$f(x)=1$$ otherwise; $$f(x)=x$$; and $$f(x)=x(x+1)\cdots (x+a-1)$$ with some integer $$a\geq 1$$. In the latter case, by some “heroic heuristics” the problem is reduced to solving a simple differential equation. Using the same argument, the case $$f(x)=x^ 2$$ is also studied. Finally, a slightly modified problem is investigated: Instead of appearing one by one in random order, in each step only the best and the worst of the remaining candidates are assumed to show with probability 1/2 each.

##### MSC:
 62L15 Optimal stopping in statistics 60G40 Stopping times; optimal stopping problems; gambling theory
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##### References:
 [1] Chow Y.S., Israel Jour. Math. 2 pp 81– (1964) · Zbl 0149.14402 · doi:10.1007/BF02759948 [2] Chow Y.S., Great Expectations (1971) [3] Mucci A.G., Ann. Prob. 1 pp 417– (1973) · Zbl 0261.60036 · doi:10.1214/aop/1176996936 [4] An extensive bibliography is given in. [5] Ferguson T.S., Statistical Science 4 pp 282– (1989) · doi:10.1214/ss/1177012493 [6] There is a large and growing literature on the secretary problem in its various modifications. Two papers not mentioned in [4] are. [7] Dhariyal I.D., Journ. Am. Statist. Assn. 76 pp 952– (1981) · doi:10.1080/01621459.1981.10477748 [8] Rubin H., Ann. Of Prob. 5 pp 627– (1977) · Zbl 0381.60036 · doi:10.1214/aop/1176995774
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