I have an experimentally observed distribution that looks very similar to a gamma or lognormal distribution. I've read that the lognormal distribution is the maximum entropy probability distribution for a random variate $X$ for which the mean and variance of $\ln(X)$ are fixed. Does the gamma distribution have any similar properties?
Gamma vs Lognormal Distributions – Key Differences
density functiongamma distributionlognormal distribution
Best Answer
As for qualitative differences, the lognormal and gamma are, as you say, quite similar.
Indeed, in practice they're often used to model the same phenomena (some people will use a gamma where others use a lognormal). They are both, for example, constant-coefficient-of-variation models (the CV for the lognormal is $\sqrt{e^{\sigma^2} -1}$, for the gamma it's $1/\sqrt \alpha$).
[How can it be constant if it depends on a parameter, you ask? It applies when you model the scale (location for the log scale); for the lognormal, the $\mu$ parameter acts as the log of a scale parameter, while for the gamma, the scale is the parameter that isn't the shape parameter (or its reciprocal if you use the shape-rate parameterization). I'll call the scale parameter for the gamma distribution $\beta$. Gamma GLMs model the mean ($\mu=\alpha\beta$) while holding $\alpha$ constant; in that case $\mu$ is also a scale parameter. A model with varying $\mu$ and constant $\alpha$ or $\sigma$ respectively will have constant CV.]
You might find it instructive to look at the density of their logs, which often shows a very clear difference.
The log of a lognormal random variable is ... normal. It's symmetric.
The log of a gamma random variable is left-skew. Depending on the value of the shape parameter, it may be quite skew or nearly symmetric.
Here's an example, with both lognormal and gamma having mean 1 and variance 1/4. The top plot shows the densities (gamma in green, lognormal in blue), and the lower one shows the densities of the logs:
(Plotting the log of the density of the logs is also useful. That is, taking a log-scale on the y-axis above)
This difference implies that the gamma has more of a tail on the left, and less of a tail on the right; the far right tail of the lognormal is heavier and its left tail lighter. And indeed, if you look at the skewness, of the lognormal and gamma, for a given coefficient of variation, the lognormal is more right skew ($\text{CV}^3+3\text{CV}$) than the gamma ($2\text{CV}$).