# Bootstrap estimate of $min_{i,j}(|X_i - X_j|)$

I have $$n$$ IID samples $$X_i$$ from some distribution. I can calculate $$\min_{i \neq j}(|X_i - X_j|)$$ and would like to get a confidence interval for said quantity.

Naive Bootstrap clearly makes no sense here. Let

$$X^{\star}_1,\dots, X^{\star}_n$$ a bootsrapped sample. Then, with high probability, there exist $$k\neq l$$ such that $$X^{\star}_k =X^{\star}_l$$ and then

$$\min_{i\neq j}(|X^{\star}_i-X^{\star}_j|) = 0$$

Is there a modification of the Bootstrap that handles this case except for the Jackknife? Is it OK to discard identical samples?

PS - I am not really interested in this statistic, I just use it to present a problem I am facing, where resampling the same measurements ruins the statistic.

• If the distributions are continuous, $\min_{i \neq j}$ is equivalent to $\min_{x_i \neq x_j}$, which would justify the "discard identical samples" approach. Oct 16, 2022 at 18:09
• Here is a Python gist demoing the use of (leave-one-out) jackknifing and dropping duplicates Oct 16, 2022 at 19:14