Abtract: Possibility of early stopping or interim sample size re-estimation lead random sample sizes. If these interim adaptations are informative, the sample size becomes a part of a sufficient statistic. Consequently, statistical inference based solely on the observed sample or the likelihood function does not use all available statistical evidence. In this work, we quantify the loss of statistical evidence using (expected) Fisher Information (FI) because observed Fisher information as a function of the likelihood does not capture the loss of statistical evidence. We decompose the total FI into the sum of the design FI and a conditional on design FI. Further, the conditional on design FI is represented as a weighted linear combination of FI conditional on realized decisions. The decomposition of total FI is useful for making a few practically useful conclusions for designing sequential experiments. In addition, this FI decomposition is used to derive a sequential version of the Cramer-Rao Lower Bound (CRLB) for estimators' mean squared errors. For a given sequential design, when the data are generated from one-parameter exponential family with canonical parameterization, the sequential CRLB is attained. Theoretical results are illustrated with a simple normal case of a two-stage design with a possibility of early stopping.
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