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

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.