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?

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?

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?

What methods can I use to determine the true PDF of my data?

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?