bpet
Oct5-09, 10:39 AM
A colleague of mine noticed that samples from a lognormal distribution seem to have much smaller kurtosis than the theoretical value - for example with \sigma=2 the theoretical value is about 9,000,000 whereas for samples of size N=1000 we found that the sample kurtosis averages around 280 and is never larger than 1000. Even with N=10^6 the sample kurtosis is only around 50,000.
We used the standard formula
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}
(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?
We used the standard formula
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}
(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?