I know mean and variance of a skewed positive random variable $X$ analytically. in literature both gamma and log-normal distributions can be fitted to such a random variable. I know that to find the best fitted distribution, maximum likelihood estimators can be used. but for that, I must run monte carlo simulations. I can estimate gamma or log-normal distributions by mean and variance which I know analytically. By monte carlo, I know that these estimated distributions, have enough accuracy and gamma distribution is better fitted.
Actually, I need 95th percentile of the random variable. I can estimate it by fitted distributions (gamma or log-normal). I prefer to use the distribution with larger 95th percentile (because I need to study the worst case, and I do not want to use MCS. I must study large numbers of different situations and perform MCS for each, but I can calculate mean and variance of each one analytically.)
I can prove that kurtosis of estimated gamma distribution is smaller than estimated log-normal distribution if both are estimated using known variance and mean. also, MCS shows that 95th percentile of estimated gamma distribution is more accurate and is larger than estimated log-normal distribution.
Is it possible to prove that 95th percentile of estimated gamma distribution is larger than estimated log-normal where mean and variance of both are the same (kurtosis can help?)?