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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.

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    $\begingroup$ 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. $\endgroup$
    – jbowman
    Oct 16, 2022 at 18:09
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    $\begingroup$ Here is a Python gist demoing the use of (leave-one-out) jackknifing and dropping duplicates $\endgroup$
    – Galen
    Oct 16, 2022 at 19:14

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