Bias approximation has played an important rôle in statistical inference, and numerous bias calculation techniques have been proposed under different contexts. We provide a unified approach to approximating the bias of the maximum likelihood estimator and the l₂ penalized likelihood estimator for both linear and nonlinear models, where the design variables are allowed to be random and the sample size can be a stopping time. The proposed method is based on the Woodroofe–Stein identity and is justified by very weak approximations. The accuracy of the derived bias formulas is assessed by simulation for several examples. The bias of the ridge estimator in high-dimensional settings is also discussed.

中文翻译：

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基于似然的估计量的偏差近似

偏差近似在统计推断中发挥了重要作用，并且在不同的背景下提出了许多偏差计算技术。我们提供了一种统一的方法来近似线性和非线性模型的最大似然估计量和l ₂惩罚似然估计量的偏差，其中允许设计变量是随机的并且样本量可以是停止时间。所提出的方法基于 Woodroofe-Stein 恒等式，并通过非常弱的近似值来证明。导出的偏差公式的准确性通过对几个示例的模拟进行评估。还讨论了高维设置中脊估计器的偏差。