The example I gave was to compare a gamma and a pareto distribution, each of which has mean 10,000 and a CV of 150% (making the standard deviation 15,000). I will spare all of you the algebra, but suffice to say, that using the Klugman-Panjer-Wilmot parameterization (which is used by most casualty actuaries in the past 20 years or so) the parameters of the gamma would be theta (R's scale) = 22500 and alpha (R's shape) = 4/9. The equivalent pareto would have theta (R's scale) = 26000 and alpha (R's shape) = 3.6.

Graphing the two (and Hadley, please forgive me for using default R' plotting, I left my ggplot book in the office;

*mea culpa*) you can easily see how the distributions are rather different.

To make things easier for me, I used the actuar package to do the graphing:

Created by Pretty R at inside-R.org

Obviously, the tails of the distributions, and thus the survival function at a given loss size, is different for the two, notwithstanding their sharing identical first two moments. So, this was just a brief but effective visualization as to how the first two moments do not contain all the information needed to find a "best fit," and why we like to use distributional fitting methods (maximum likelihood, maximum spacing, various minimum distance metrics like Cramer-von Mises, etc.) to get a better understanding of the potential underlying loss processes.

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