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Assume I have a sample of real data and from which I create a set of new samples with replacement. I next compute the means for each 'synthetic' sample I generated.

My first question is as follows. I presume the central limit theorem applies and the distribution of these means will be normal, or at least approach normal as the size of the number of samples generated gets bigger. Is that true?

If this is true then I assume the 2.5 and 97.5 percentiles of the distribution of the means will be symmetrical even though the original distribution from which the synthetic samples were generated was not normal? Is that true?

I'm a bit confused on these points as I thought the confidence interval would be asymmetric if the underlying distribution was also asymmetric. I've done simulations and I do seem to get asymmetric intervals but this could be because I'm not sampling enough (I generated 500,000 samples) and as suggested above the central limit theroem suggests the distribution should be normal.

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To your first question, yes, this is what CLT tells us given that the sample observations are also independent and bare approximately equal effect.

To your second question, it is true only given enough generated samples. If the underlying data distribution is not symmetric, you can only determine it after a lengthy simulation that you need n samples to ensure the Gaussian shape. You can get an asymmetric observed statistic distribution if you data are very skewed or biased (heavy tailed).

You can be guaranteed that your statistic distribution will be Gaussian if the number of samples approaches infinity with any underlying distribution. But with limited number of simulations your estimate of quantiles bay be biased.

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  • $\begingroup$ Thanks for that answer. Is it then true that to get reliable confidence intervals using a bootstrap we need to do many resamplings? I see people using 1000 resampling but that seem far too small. $\endgroup$
    – rhody
    Nov 2, 2017 at 14:20
  • $\begingroup$ @rhody, you are right. I am not apt at numeric methods, but my senior statistician colleague used at least 20K samplings to estimate the density, and as a matter of fact even after that lengthy thing the density was often skewed. It is good for you to read special publications on this topic. $\endgroup$
    – Alexey Burnakov
    Nov 2, 2017 at 14:25

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