Delete-d jackknife

Instead of leaving out one observation at a time, we leave out d observations. Therefore, the size of a delete-d jackknife sample is (n - d), and there are $\binom{n}{d}$ jackknife samples.

Let $\hat{\theta}_{(s)}$ denote $\hat{\theta}$ applied to the data set with subset s removed. The formula for the delete-d jackknife estimate of s.e. is

$$\{ \frac{n - d}{d \binom{n}{d}} \sum (\hat{\theta}_{(s)} - \hat{\theta}_{(\cdot)})^2 \}^{1/2}$$ (4)

where $\hat{\theta}_{(\cdot)} = \sum \hat{\theta}_{(s)} / \binom{n}{d}$ and the sum is over all subsets s of size (n - d) chosen without replacement for $x_1, x_2, \ldots x_n$.

It can be shown that the delete-d jackknife is consistent for the median if $\sqrt{n} / d \rightarrow 0$ and $(n-d) \rightarrow \infty$. Roughly speaking, it is preferrable to choose a d such that $\sqrt{n} < d < n$ for the delete-d jackknife estimation of standard error.