Chapter II
Extreme Events
16
Figure II-1 Representative schema of the block maxima and POT methods.
II.2.1. Block maxima method
Let 𝑋 1 , 𝑋 2 , ⋯ , 𝑋 𝑛 be a sequence of independent distributed random variables, where 𝑋 𝑖 , 𝑖 =
1 … , 𝑛 has some unknown distribution function 𝐹 . Define 𝑀𝑛 = 𝑚𝑎𝑥{𝑋 1 , 𝑋 2 , ⋯ , 𝑋 𝑛 }, i.e.,
𝑀 𝑛 is the block maxima. The objective is now to find the distribution of 𝑀 𝑛 . Since 𝑋 has
distribution function 𝐹 (Andersson, K. 2020), we could argue that:
Equation II-1
This is insufficient since 𝐹 is unknown, (Coles, 2001). Fortunately, the Fisher-Tippet theorem
can be employed in this case. This theorem states that as 𝑛 approaches infinity, 𝑀ₙ converges in
distribution to the generalized extreme value (GEV) family of distributions (Andersson, K.
2020). These distributions are defined in the Equation II-2
Equation II-2
Provided that 1 + 𝜉(𝑥 − µ) = 𝜎 > 0. The three parameters µ, 𝜎 and 𝜉 are the location parameter,
the scale parameter and the shape parameter respectively. µ indicates where on the axis the
distribution of 𝑀 𝑛 is located and 𝜎 illustrate the width of the distribution. Finally, 𝜉 is a measure
of the shape of the distribution.
• 𝜉 > 0, then 𝑀 𝑛 follows the Fréchet distribution, which is a distribution with heavy tails.
• 𝜉 = 0, 𝑀 𝑛 follows the Gumbel distribution, which has exponential tails. For 𝜉 < 0. The
tails of this distribution are lighter than in the normal one.
• 𝜉 < 0, the distribution has an upper end-point; µ − 𝜎 = 𝜉