# Interesting lognormal kurtosis discrepancy

1. Oct 5, 2009

### bpet

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?

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