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：我们考虑在估计量存在（渐近）偏误情况下的自举推断。研究表明，即使偏误项无法被一致估计，通过适当实施自举方法仍可获得有效推断。具体而言，我们证明了Beran(1987, 1988)最初为提供高阶修正而提出的预枢轴方法，通过将原始自举p值转化为渐近均匀随机变量，恢复了自举的有效性。我们提出了预枢轴的两种不同实施方式（插入法与双重自举法），并给出了确保自举推断有效性的通用高阶条件。为阐明研究结果的实际相关性与实施方法，我们讨论五个实例：(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)无限方差数据的位置模型；以及(v)动态面板数据模型。