Adaptive Linear Estimating Equations
This work addresses statistical inference challenges in sequential data collection, such as in multi-armed bandits, by connecting non-asymptotic and asymptotic inference paradigms, though it appears incremental as it builds on existing methods.
The paper tackles the problem of non-normal asymptotic behavior in ordinary least squares estimators for adaptive linear regression models, proposing a debiased estimator based on adaptive linear estimating equations that achieves asymptotic normality and near-optimal asymptotic variance.
Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality.