TY - JOUR

T1 - Optimal Stopping for Interval Estimation in Bernoulli Trials

AU - Yaacoub, Tony

AU - Moustakides, George V.

AU - Mei, Yajun

N1 - Funding Information:
Manuscript received November 18, 2017; revised August 3, 2018; accepted November 24, 2018. Date of publication December 6, 2018; date of current version April 19, 2019. This work was supported in part by the U.S. National Science Foundation through Rutgers University under Grant CIF 1513373 and in part by the U.S. National Science Foundation through the Georgia Institute of Technology under Grants CMMI 1362876 and DMS 1830344.
Funding Information:
This work was supported in part by the U.S. National Science Foundation through Rutgers University under Grant CIF 1513373 and in part by the U.S. National Science Foundation through the Georgia Institute of Technology under Grants CMMI1362876 and DMS 1830344.
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2019/5

Y1 - 2019/5

N2 - We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion \theta. Assuming that an independent and identically distributed sequence of Bernoulli ( \theta ) trials is observed sequentially, we are interested in designing: 1) a stopping time T that will decide the best time to stop sampling the process and 2) an optimum estimator \hat{{\theta}}-{{T}} that will provide the optimum center of the interval estimate of \theta. We follow a semi-Bayesian approach, where we assume that there exists a prior distribution for \theta , and our goal is to minimize the average number of samples while we guarantee a minimal specified coverage probability level. The solution is obtained by applying standard optimal stopping theory and computing the optimum pair (T,\hat{{\theta }}-{{T}}) numerically. Regarding the optimum stopping time component T , we demonstrate that it enjoys certain very interesting characteristics not commonly encountered in solutions of other classical optimal stopping problems. In particular, we prove that, for a particular prior (beta density), the optimum stopping time is always bounded from above and below; it needs to first accumulate a sufficient amount of information before deciding whether or not to stop, and it will always terminate before some finite deterministic time. We also conjecture that these properties are present with any prior. Finally, we compare our method with the optimum fixed-sample-size procedure as well as with existing alternative sequential schemes.

AB - We propose an optimal sequential methodology for obtaining confidence intervals for a binomial proportion \theta. Assuming that an independent and identically distributed sequence of Bernoulli ( \theta ) trials is observed sequentially, we are interested in designing: 1) a stopping time T that will decide the best time to stop sampling the process and 2) an optimum estimator \hat{{\theta}}-{{T}} that will provide the optimum center of the interval estimate of \theta. We follow a semi-Bayesian approach, where we assume that there exists a prior distribution for \theta , and our goal is to minimize the average number of samples while we guarantee a minimal specified coverage probability level. The solution is obtained by applying standard optimal stopping theory and computing the optimum pair (T,\hat{{\theta }}-{{T}}) numerically. Regarding the optimum stopping time component T , we demonstrate that it enjoys certain very interesting characteristics not commonly encountered in solutions of other classical optimal stopping problems. In particular, we prove that, for a particular prior (beta density), the optimum stopping time is always bounded from above and below; it needs to first accumulate a sufficient amount of information before deciding whether or not to stop, and it will always terminate before some finite deterministic time. We also conjecture that these properties are present with any prior. Finally, we compare our method with the optimum fixed-sample-size procedure as well as with existing alternative sequential schemes.

KW - Sequential estimation

KW - binomial proportion

KW - confidence intervals

KW - optimal stopping

KW - sequential analysis

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U2 - 10.1109/TIT.2018.2885405

DO - 10.1109/TIT.2018.2885405

M3 - Article

AN - SCOPUS:85058085029

SN - 0018-9448

VL - 65

SP - 3022

EP - 3033

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 5

M1 - 8565954

ER -