自助法重抽样的理解; Understanding the Residual Bootstrap 作者: bluish 时间: 2026-02-08 分类: 笔记 # 自助法重抽样 设想数据：y ～ beta * x1 + residual 我们的目标是通过bootstrap获得beta的置信区间（即其估计波动）。 主要思想： 1. 在Bootstrap中，我们将original sample视作反映population distribution的empirical population，并根据其的resample下的估计量的波动来反映original sample的估计量的置信区间。 2. 在固定x1与beta_hat的条件下，empirical population（即original sample）的分布是由residual来体现的；因此，bootstrap sample（即resample得到的sample）的分布或随机性也是由residuals来决定。 3. 因此，在对residual进行一轮bootstrap后，将residuals加回beta_hat * x1，就体现了empirical population下的一轮bootstrap sample的样本分布。 4. 进行多轮residual bootstrap后，对每一轮的bootstrap sample重新进行OLS拟合，就能从所有轮次得到的beta的估计值中摘得置信区间。 # Residual Bootstrap Suppose the data: y ～ beta * x1 + residual Our goal is to use bootstrap to obtain the confidence interval of beta (i.e., the sampling variability of its estimator). Main idea: 1. In bootstrap, we treat the original sample as an empirical population that reflects the population distribution, and we approximate the confidence interval of the estimator of the original sample by the variability of the estimator under resampling. 2. Conditional on fixed x1 and beta_hat, the distribution of the empirical population (i.e., the original sample) is captured by the residuals; therefore, the distribution or randomness of the bootstrap samples (i.e., resampled samples) is also driven by the residuals. 3. Consequently, after performing one round of bootstrap on the residuals, adding the bootstrapped residuals back to beta_hat * x1 yields one bootstrap distrubition drawn from the empiriacal population. 4. By repeating the residual bootstrap many times and refitting OLS on each bootstrap sample, we can construct the confidence interval for beta from the set of resulting bootstrap estimates. 自助法重抽样的理解; Understanding the Residual Bootstrap https://bluish.net/archives/2360/ 作者 bluish 发布时间 2026-02-08 许可协议 CC BY-SA 4.0 复制版权信息 标签: none