Sergey Tarima : Cost of Sequential Adaptations and Lower Bound on Mean Squared Error of Post-Adaptation Estimators

Posted by Min Yang , part of the Statistics and Data Science Seminar.

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Sept. 13, 2023, 4 p.m.

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636 SEO

Abstract

The possibility of early stopping and/or interim sample size re-estimation lead to random sample sizes. When such interim adaptations are informative, the interim decision becomes a component of the sufficient statistic. We decompose the total Fisher Information (FI) into the design FI and a conditional-on-design FI analogous to Molenberghs et al. (2014). We go further, representing the conditional-on-design FI as a weighted linear combination of FIs conditional on realized decisions. This decomposition is useful for quantifying how much mean-squared error will be lost due to planned-informative adaptations. We use The FI unspent by having a planned-informative adaptation to determine the lower bound on mean squared error of post-adaptation estimators [the Cramer-Rao lower bound (1946) and its sequential version suggested by Wolfowitz (1947) are not applicable to such estimators]. Theoretical results are illustrated with simple normal samples collected according to a two-stage design with a possibility of early stopping.