TERStatistics of Extremes.pdf


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1.5

Quantile-Quantile Plot

The central limit theorem is generally used in statistical applications by interpreting
(1.1) as an approximation for the distribution of the sample mean X for large n.
This theorem is helpful for applications because the approximating distribution of
the sample mean is normal regardless of the parent population of the Xi . Similar
arguments will be use in this paper to obtain approximating distributions for sample
extremes.

1.5

Quantile-Quantile Plot

The Quantile–Quantile(Q-Q) plot is a graph for determining if two data sets come from
populations with a common distribution. This is a plot of the quantiles of the first data
set against the quantiles of the second data set. So it compares empirical quantiles
with theoretical quantiles. This plot should be linear (or at least near diagonal) if the
adjustment is good.

2
2.1

Basics of the Univariate Extreme Value Theory
Limit Distribution of the Maximum

The model focuses on the statistical behavior of Mn . We know that the distribution
of Mn is given by:
FMn = P(Mn ≤ x)
= P(

n
\

(Xi ≤ x))

i=1

=
=

n
Y
i=1
n
Y

P(Xi ≤ x)
F (x) = F n (x)

∀x ∈ IR

i=1

But it is not helpful in practice, since the distribution function F is generally unknown.
An other nonparametric approach would be to use the empirical function distribution
6