The Impact of Application of the Jackknife to the Sample Median

The jackknife is a reliable tool for reducing the bias of a wide range of estimators. This note demonstrates that even such versatile tools have regularity conditions that can be violated even in relatively simple cases, and that caution needs to be exercised in their use. In particular, we show that the jackknife does not provide the expected reliability for bias-reduction for the sample median, because of subtle changes in behavior of the sample median as one moves between even and odd sample sizes. These considerations arose out of class discussions in a MS-level nonparametrics course.

刀切法（jackknife）是用于降低各类估计量偏差的可靠工具。本研究表明，即便这类通用性极强的工具，也存在正则性条件，且即便在相对简单的场景中也可能被违反，因此使用时需保持谨慎。具体而言，当样本量在奇数与偶数之间变动时，样本中位数的统计性质会发生细微变化，因此刀切法无法在样本中位数的偏差缩减任务中实现预期的可靠性。上述思考源自一门硕士层次非参数统计课程的课堂讨论。