# A sharp bound for winning within a proportion of the maximum of a sequence

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## Abstract

This note considers a variation of the full-information secretary problem where the random variables to be observed are continuous and iid. Consider the independent sequence of continuous random variables \(X_1,\ldots ,X_n\), \(M_n:=\max \{X_1,\ldots ,X_n\}\), and the objective to select the maximum of the sequence. What is the maximum probability of “stopping at the maximum”? That is, what is the stopping time \(\tau \) adapted to \(X_1,\ldots ,X_n\) that maximizes \(P(X_{\tau }=M_n)\)? This problem was examined by Gilbert and Mosteller (J Am Stat Assoc 61:35–73, 1966) and the optimal win probability in this case is denoted by \(v_{n,\mathrm{max}}^*\). What if it is desired to “stop within a proportion of the maximum”? That is, for \(0<\alpha <1\), what is the stopping rule \(\tau \) that maximizes \(P(X_{\tau } \ge \alpha M_n)\)? In this note it is proven that for any continuous random variable *X* if \(\tau ^*\) is the optimal stopping rule then \(P(X_{\tau ^*} \ge \alpha M_n)\ge v_{n,\mathrm{max}}^*\), and this lower bound is sharp. Some examples and another interesting result are presented.

## Keywords

Choosing the maximum stopping time Sharp inequalities for stochastic processes## Mathematics Subject Classification

Primary 60G40## Notes

### Acknowledgements

To Pieter Allaart for his comments and suggestions in this project.

## References

- 1.Ferguson, T.S.: Who solved the secretary problem? (With comments and a rejoinder by the author.). Stat. Sci.
**4**(3), 282–296 (1989)CrossRefGoogle Scholar - 2.Gilbert, J.P., Mosteller, F.: Recognizing the maximum of a sequence. J. Am. Stat. Assoc.
**61**, 35–73 (1966)MathSciNetCrossRefGoogle Scholar