Comparison of Stopping Rules in Sequential Estimation of the Number of Classes in a Population

Abstract

In sequential analysis, investigation of stopping rules is important, as they govern the sampling cost and derivation and accuracy of frequentist inference. We study stopping rules in sampling from a population comprised of an unknown number of classes where all classes are equally likely to occur in each selection. We adopt Blackwell's criterion for a “more informative experiment” to compare stopping rules in our context and derive certain complete class results, which provide some guidance for selecting a stopping rule. We show that it suffices to let the stopping probability, at any time, depend only on the number of selections and the number of discovered classes up to that time. A more informative stopping rule costs a higher expected sample size, and conversely, any given stopping rule can be improved with an increment in expected sample size. Admissibility within all stopping rules with a uniform upper bound on average sample size is also discussed. Any fixed-sample-size rule is shown to be admissible within an appropriate class. Finally, we show that for the minimal sufficient statistic to be complete, which is useful for unbiased estimation, the stopping rule must be nonrandomized.

Keywords:

• Admissibility
• Capture–recapture
• Completeness
• Sufficiency
• Unbiasedness

Subject Classification:

• 62L10
• 62C07
• 62B15

ACKNOWLEDGMENTS

The authors sincerely thank an anonymous referee, an Associate Editor, and the Editor for carefully reading the manuscript and providing many helpful comments and suggestions.

Notes

Recommended by Bennett Eisenberg