Timeline for Biased bootstrap: is it okay to center the CI around the observed statistic?
Current License: CC BY-SA 3.0
10 events
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| Nov 29, 2015 at 9:19 | history | edited | NRH | CC BY-SA 3.0 |
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| Jun 27, 2015 at 17:25 | history | bounty ended | CommunityBot | ||
| Jun 26, 2015 at 7:14 | comment | added | NRH | Sorry ZNK, I misunderstood your question. If you increase the sample size $n$, the bias will be smaller, yes! The estimator is consistent. Precisely for the uniform distribution I would be somewhat sceptical about the actual coverage of the confidence intervals even for large $n$ for the reasons I described in the answer. For all other distributions the CLT applies, and the different methods will produce asymptotically correct coverage for $n \to \infty$. | |
| Jun 25, 2015 at 23:51 | comment | added | ZNK |
To be clear, I'm not talking about increasing the number of bootstraps, but rather increasing the sample size per bootstrap replicate, so instead of drawing 100 samples with replacement, it draws 10000 samples with replacement. When I use this approach, the bias is about -0.04 and the confidence intervals encapsulate the data, but never go over 3.912. This is the function I used: my_sampler <- function(x, mle = NULL){n <- length(x); rmultinom(1, n*n, tabulate(x)/n)}. Are the CIs obtained from this method just as silly and meaningless?
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| Jun 25, 2015 at 21:23 | comment | added | NRH | Increasing the number of bootstrap samples will not really help. It has to be large enough so that you can reliable estimate the quantities of interest for the distribution of $\theta(\mathbf{p}_n^*)$, say, but otherwise increasing the number of bootstrap samples will not remove the bias or make the confidence any more appropriate. | |
| Jun 25, 2015 at 18:22 | comment | added | ZNK | I apologize if I come off as a bit naive, but I noticed that if I set up the bootstrap to sample $n^2$ samples from a multinomial distribution where the probabilities are the observed proportions, the bias and the CI was reduced. Would increasing the number of samples used for the bootstrap be a method for addressing this? | |
| Jun 25, 2015 at 12:45 | comment | added | EdM | I didn't know this literature either, until this question and your answer came up. Which is somewhat embarrassing, since Shannon entropy is often used as a measure in my area of biomedical science. I'll see what I can put together as an additional answer. | |
| Jun 25, 2015 at 7:17 | comment | added | NRH | @EdM this is very useful information. I did not know the literature on this particular bias problem. It could be really useful if you could turn the comment into an answer that explains the bias correction and how it could be used with bootstrapping, say, to obtain confidence intervals. | |
| Jun 24, 2015 at 20:56 | comment | added | EdM | The bias problem with using the "plug-in" estimator for entropy has been appreciated for decades. This paper analyzes less-biased estimates. A bias correction up to order $1/n$, which dates to 1955 (see eq. 4 of the linked paper), can be applied to the case presented by the OP. The correction is 0.245, almost identical to the bias identified by the bootstrap. Perhaps the bootstrap should be used here for estimating the entropy itself, not just its confidence limits. | |
| Jun 24, 2015 at 19:25 | history | answered | NRH | CC BY-SA 3.0 |