An accurate percentile method for parametric inference based on asymptotically biased estimators

TL;DR

This paper introduces the implicit bootstrap, a simulation-based method for constructing valid confidence intervals even with biased estimators, especially useful in complex parametric settings with data issues.

Contribution

It proposes a novel implicit bootstrap approach that remains valid despite asymptotic bias, improving confidence interval accuracy in challenging parametric inference scenarios.

Findings

The implicit bootstrap provides asymptotically valid confidence intervals.

It achieves second-order accuracy in coverage.

The method is exact in cases where standard bootstrap fails.

Abstract

Inference methods for computing confidence intervals in parametric settings usually rely on consistent estimators of the parameter of interest. However, it may be computationally and/or analytically burdensome to obtain such estimators in various parametric settings, for example when the data exhibit certain features such as censoring, misclassification errors or outliers. To address these challenges, we propose a simulation-based inferential method, called the implicit bootstrap, that remains valid regardless of the potential asymptotic bias of the estimator on which the method is based. We demonstrate that this method allows for the construction of asymptotically valid percentile confidence intervals of the parameter of interest. Additionally, we show that these confidence intervals can also achieve second-order accuracy. We also show that the method is exact in three instances where the standard bootstrap fails. Using simulation studies, we illustrate the coverage accuracy of the method in three examples where standard parametric bootstrap procedures are computationally intensive and less accurate in finite samples.