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I am taking a stats course right now and we're studying the bootstrap. One lecture slide says:

"These methods for creating a confidence interval only work if the bootstrap distribution is smooth and symmetric"

If the bootstrap distribution is highly skewed or looks “spiky” with gaps, you will need to go beyond intro stat to create a confidence interval

I didn't get an explanation on why this is so. Why do we need a smooth and symmetric bootstrap?

EDIT:

We're simply bootstrapping the mean statistic. We compute N bootstrap datasets from the given sample dataset then compute the mean of each bootstrap dataset and visualize the resulting distribution. That is when I was told the above.

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    $\begingroup$ What is the confidence interval that you've learned? Is it of the form $\hat \theta \pm 2 \hat{SE}^*(\hat \theta)$ where $\hat{SE}^*$ comes from the bootstrap distribution? $\endgroup$ – jld Mar 28 '16 at 0:26
  • $\begingroup$ It is just the standard "bootstrap procedure". Sorry if that is unclear, I thought the bootstrap procedure was the same throughout stats $\endgroup$ – user46925 Mar 28 '16 at 1:42
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    $\begingroup$ Regrettably not. The principle is the same, but there are many possible bootstraps, depending e.g. on how much of the model you trust, what sorts of dependencies there are in the data, and how well behaved the statistic is that you are bootstrapping. $\endgroup$ – conjugateprior Mar 28 '16 at 2:05
  • $\begingroup$ Thanks guys. I've updated the question - please let me know if you need any other updates $\endgroup$ – user46925 Mar 28 '16 at 2:11
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    $\begingroup$ Recall that with parametric/asymptotic assumptions you are typically assuming that the sampling distribution of interest takes some smooth functional form, e.g. Normal, and with the bootstrap you are are assuming that this distribution function is reasonably well represented by the empirical distribution function from the data you have. $\endgroup$ – conjugateprior Mar 28 '16 at 2:42

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