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All of the jackknife methods (JN) I have seen (for example) use the JN to estimate standard errors and then use those estimates in standard normal-assumption symmetric intervals constructions ($ \hat{\mu} \pm z_{1 - \alpha} \cdot \hat{se} $),

Can the jackknife be used to estimate the sample statistic distribution directly ala the bootstrap? In other words, can the values returned from jackknife replication be used as inputs to percentile or BCa-type confidence interval functions to get asymmetric interval? Can Jackknife only be used to find standard errors?

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Yes, it can. Example: https://influentialpoints.com/Training/log_normal_confidence_intervals.htm

Say you have lognormal-distributed data (this is right-skewed and positive), and you're trying to estimate the mean $D$. If you use a normal assumption, you could get a nonsensical CI which includes negative values. If you use a log-normal assumption, you would still calculate the standard error using the jacknife, but, the way you apply it to generate a CI is different. In the case in the above link, the jacknife SE plugs into a formula for a value called $C$, which is basically a log-scale confidence interval width.

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    $\begingroup$ It can give asymmetric confidence intervals when we replace the standard normal assumption, but can it also do the second part of the question "Can the jackknife be used to estimate the sample statistic distribution directly ala the bootstrap?" $\endgroup$
    – Sextus Empiricus
    Jan 19 at 11:23

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