Normalized centered moments of the Fréchet extreme-value distribution and inference of its parameter

In the present work, we provide the general expression of the normalized centered moments of the Fr\'echet extreme-value distribution. In order to try to represent a set of data corresponding to rare events by a Fr\'echet distribution, it is important to be able to determine its characteristic parameter $\alpha$. Such a parameter can be deduced from the variance (proportional to the square of the Full Width at Half Maximum) of the studied distribution. However, the corresponding equation requires a numerical resolution. We propose two simple estimates of $\alpha$ from the knowledge of the variance, based on the Laurent series of the Gamma function. The most accurate expression involves the Ap\'ery constant.