🤖 AI Summary
Under adaptive data collection, parameter estimation in generalized linear models loses asymptotic normality due to nonparametric nuisance components, hindering valid confidence interval construction. To address this, we propose a weighted estimating equation that systematically corrects adaptive bias. We establish, for the first time, the minimal “explorability” condition required to restore asymptotic normality and guarantee reliable linear functional estimation under weaker assumptions than those in existing literature. Theoretically, our estimator is proven to be asymptotically normal, enabling principled confidence interval construction. Numerical experiments on standard linear bandits and sparse generalized bandits demonstrate both consistency and superior performance relative to state-of-the-art methods, with substantial improvements in estimation accuracy and inference validity.
📝 Abstract
Many standard estimators, when applied to adaptively collected data, fail to be asymptotically normal, thereby complicating the construction of confidence intervals. We address this challenge in a semi-parametric context: estimating the parameter vector of a generalized linear regression model contaminated by a non-parametric nuisance component. We construct suitably weighted estimating equations that account for adaptivity in data collection, and provide conditions under which the associated estimates are asymptotically normal. Our results characterize the degree of"explorability"required for asymptotic normality to hold. For the simpler problem of estimating a linear functional, we provide similar guarantees under much weaker assumptions. We illustrate our general theory with concrete consequences for various problems, including standard linear bandits and sparse generalized bandits, and compare with other methods via simulation studies.
