Interesting lognormal kurtosis discrepancy

[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?

 

[mmwave]

Thank you for bringing this up, colleague. The issue you have observed is actually quite common and can be explained by the slow convergence rate of the sample kurtosis to the theoretical value. This is due to the fact that the kurtosis is a higher order moment and is more sensitive to extreme values in the data.

To understand the convergence rate, we can look at the central limit theorem. This theorem states that as the sample size (N) increases, the sampling distribution of the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. However, this theorem does not hold for higher order moments such as the kurtosis.

In fact, the convergence rate of the sample kurtosis is much slower than that of the sample mean. It has been shown that the convergence rate of the sample kurtosis is on the order of 1/\sqrt{N}, which means that as the sample size increases, the sample kurtosis will approach the theoretical value at a much slower rate compared to the sample mean.

This slow convergence rate is also affected by the shape of the underlying distribution. In the case of a lognormal distribution, the kurtosis is already high (around 9,000,000 for \sigma=2), so it will take a much larger sample size to see the sample kurtosis approach this value. This is why even with a sample size of 10^6, the sample kurtosis is only around 50,000.

In summary, the slow convergence rate of the sample kurtosis is a well-known phenomenon and is due to the higher order nature of this moment and the shape of the underlying distribution. It is important to keep this in mind when interpreting kurtosis values from a sample.