We consider bootstrap inference for estimators which are (asymptotically) biased. We show that, even when the bias term cannot be consistently estimated, valid inference can be obtained by proper implementations of the bootstrap. Specifically, we show that the prepivoting approach of Beran (1987, 1988), originally proposed to deliver higher-order refinements, restores bootstrap validity by transforming the original bootstrap p-value into an asymptotically uniform random variable. We propose two different implementations of prepivoting (plug-in and double bootstrap), and provide general high-level conditions that imply validity of bootstrap inference. To illustrate the practical relevance and implementation of our results, we discuss five examples: (i) inference on a target parameter based on model averaging; (ii) ridge-type regularized estimators; (iii) nonparametric regression; (iv) a location model for infinite variance data; and (v) dynamic panel data models.

翻译：我们考虑针对具有（渐近）偏差的估计量的自助法推断。研究表明，即使偏差项无法被一致估计，通过恰当实施自助法仍能获得有效推断。具体而言，Beran（1987, 1988）最初为提升高阶精度而提出的预枢轴方法，通过将原始自助法p值转化为渐近均匀随机变量，从而恢复自助法的有效性。我们提出两种不同的预枢轴实施方法（插入法与双自助法），并给出确保自助法推断有效性的通用高阶条件。为说明研究结果的实际相关性与实施过程，本文讨论五个典型案例：（i）基于模型平均的目标参数推断；（ii）岭型正则化估计量；（iii）非参数回归；（iv）无穷方差数据的位置模型；（v）动态面板数据模型。