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The cost of sequential adaptation and the lower bound for mean squared error

Author

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  • Sergey Tarima

    (Medical College of Wisconsin)

  • Nancy Flournoy

    (University of Missouri-Columbia)

Abstract

Informative interim adaptations lead to random sample sizes. The random sample size becomes a component of the sufficient statistic and estimation based solely on observed samples or on the likelihood function does not use all available statistical evidence. The total Fisher Information (FI) is decomposed into the design FI and a conditional-on-design FI. The FI unspent by a design’s informative interim adaptation decomposes further into a weighted linear combination of FIs conditional-on-stopping decisions. Then, these components are used to determine the new lower mean squared error (MSE) in post-adaptation estimation because the Cramer–Rao lower bound (1945, 1946) and its sequential version suggested by Wolfowitz (Ann Math Stat 18(2):215–230, 1947) for non-informative stopping are not applicable to post-informative-adaptation estimation. In addition, we also show that the new proposed lower boundary on the MSE is reached by the maximum likelihood estimators in designs with informative adaptations when data are coming from one-parameter exponential family. Theoretical results are illustrated with simple normal samples collected according to a two-stage design with a possibility of early stopping.

Suggested Citation

  • Sergey Tarima & Nancy Flournoy, 2024. "The cost of sequential adaptation and the lower bound for mean squared error," Statistical Papers, Springer, vol. 65(9), pages 5529-5553, December.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:9:d:10.1007_s00362-024-01565-x
    DOI: 10.1007/s00362-024-01565-x
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    References listed on IDEAS

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    1. Sergey Tarima & Nancy Flournoy, 2022. "Most powerful test sequences with early stopping options," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(4), pages 491-513, May.
    2. Nancy Flournoy & Sergey Tarima, 2023. "Posterior alternatives with informative early stopping," Statistical Papers, Springer, vol. 64(4), pages 1329-1341, August.
    3. Sergey Tarima & Nancy Flournoy, 2019. "Asymptotic properties of maximum likelihood estimators with sample size recalculation," Statistical Papers, Springer, vol. 60(2), pages 373-394, April.
    Full references (including those not matched with items on IDEAS)

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