  In this short note, I comment on the research of Pisarenko et al. (2014)
regarding the extreme value theory and statistics in case of earthquake
magnitudes. The link between the generalized extreme value distribution (GEVD)
as an asymptotic model for the block maxima of a random variable and the
generalized Pareto distribution (GPD) as a model for the peak over thresholds
(POT) of the same random variable is presented more clearly. Pisarenko et al.
(2014) have inappropriately neglected that the approximations by GEVD and GPD
work only asymptotically in most cases. This applies particularly for the
truncated exponential distribution (TED), being a popular distribution model
for earthquake magnitudes. I explain why the classical models and methods of
the extreme value theory and statistics do not work well for truncated
exponential distributions. As a consequence, the classical methods should be
used for the estimation of the upper bound magnitude and corresponding
parameters. Furthermore, different issues of statistical inference of Pisarenko
et al. (2014) are commented on and alternatives are proposed. Arguments are
presented why GPD and GEVD would work for different types of stochastic
earthquake processes in time, and not only for the homogeneous (stationery)
Poisson process as assumed by Pisarenko et al. (2014). The crucial point of
earthquake magnitudes is the poor convergence of their tail distribution to the
GPD, and not the earthquake process in time.
