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Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

Published online by Cambridge University Press:  05 October 2011

ALESSANDRO ARLOTTO  and
J. MICHAEL STEELE
ALESSANDRO ARLOTTO
Affiliation:
Department of Operations and Information Management, Wharton School, Huntsman Hall 527.2, University of Pennsylvania, Philadelphia, PA 19104, USA (e-mail: alear@wharton.upenn.edu)
J. MICHAEL STEELE
Affiliation:
Department of Statistics, Wharton School, Huntsman Hall 447, University of Pennsylvania, Philadelphia, PA 19104, USA (e-mail: steele@wharton.upenn.edu)
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Abstract

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 799 - 814
Copyright
Copyright © Cambridge University Press 2011

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