The Cost of Sequential Adaptation and the Lower Bound for Mean Squared Error

TL;DR

This paper investigates the impact of sequential adaptations on estimation accuracy, deriving lower bounds for mean squared error based on Fisher Information decomposition in adaptive sampling designs.

Contribution

It introduces a theoretical framework for quantifying the cost of interim adaptations and establishes a lower bound for estimation error considering unspent Fisher Information.

Findings

Decomposition of Fisher Information into design and conditional components.

Lower bounds for mean squared error in two-stage adaptive designs.

Illustrations with normal sample data and early stopping scenarios.

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 the interim adaptation is used to determine the lower mean squared error in post-adaptation estimation. Theoretical results are illustrated with simple normal samples collected according to a two-stage design with a possibility of early stopping.