Working with the bootstrap sample vs the original sample

Question

Consider a sample of real numbers. Say we want to estimate the central tendency of the population and get a sense of our uncertainty around this estimation.

Let's put assumptions about the population distribution aside for a moment, and consider the following two approaches.

1. Get a bootstrap sample of the input sample. That is, sample with replacement (e.g. get 100 resamples) and compute the mean for each resample. We then output the mean and confidence intervals on the resulting empirical distribution of means.
2. We output the mean from the input sample, and percentiles around the mean to convey uncertainty around the estimate.

Bootstrap vs original sample:

• While I understand what approach #1 does. Is there an underlying estimator behind #2?
• What would the percentiles around the mean in #2 convey in contrast to the CI of #1? Approach #2 conveys a sense of uncertainty, but I am having a hard time relating it to a frequentist or Bayesian interpretation.
• Would method #2 ever provide a better estimator of the population mean? (e.g. less biased and lower variance)?

Answer 1

The estimator in #2 is the thing you're generating the interval for ... the sample mean. You're using the bootstrap to try to get at the sampling distribution of the sample mean, by using the resampling distribution to approximate it.

Since it's the exact same estimator in #1 and #2, #2 will have the same true properties (whatever they are, since you don't actually know the true distribution, the true level of dependence, and so on) as in #1, you're just trying to get at one of those properties in two different ways.