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I’ve seen two ways to use bootstrapping to estimate confidence intervals of parameters estimated via maximum likelihood

The first method fits the data with the assumed distribution. Then in a loop samples from that distribution N (same size as the original data) random numbers and refits using maximum likelihood. Repeat many times. Output is a list of parameter values.

The second method fits the data with the assumed distribution. Then samples with replacement from the original data a new set of data. Same number of elements as the original data. For each of these samples, fit using maximum likelihood.

Is one way better or more theoretically correct? I was trying to figure out which way is closer to the asymoptic assumptions in frequentist theory.

Thanks for any guidance.

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