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Interesting lognormal kurtosis discrepancy

  1. Oct 5, 2009 #1
    A colleague of mine noticed that samples from a lognormal distribution seem to have much smaller kurtosis than the theoretical value - for example with [tex]\sigma=2[/tex] the theoretical value is about 9,000,000 whereas for samples of size [tex]N=1000[/tex] we found that the sample kurtosis averages around 280 and is never larger than 1000. Even with [tex]N=10^6[/tex] the sample kurtosis is only around 50,000.

    We used the standard formula
    [tex]K=\frac{E[(X-E[X])^4]}{E[(X-E[X])^2]^2}=e^{4\sigma^2}+2e^{3\sigma^2}+3e^{2\sigma^2}-3 \approx e^{4\sigma^2}[/tex]
    (for excess kurtosis subtract 3 again).

    Several things could be contributing here though I think it's to do with slow convergence to the central limit; factors such as machine precision, random number generation method and use of the unbiased estimator don't seem to make a difference in the results. Anyone come across this before or know how to derive the convergence rate?
  2. jcsd
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