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on m and n , and on the distribution function of the population from which the samples
are drawn. When the distribution function of the population as a whole is mentioned
in reference to the distribution function of the quantiles, it is often called the initial
distribution . The extreme values of the samples are their smallest and their largest items.
Obviously, the distribution functions of these extreme values of the samples depend only
on n , the number of items in the samples, and on the initial distribution. When the size
of the samples is very large, i.e. in the limit when n
, the distribution functions of
the extreme values are called asymptotic distribution functions or asymptotes . Clearly,
an extreme value asymptote no longer depends on m and n , but only on the nature of its
initial distribution.
In the study of extreme values three general types of initial distribution F ( x ) have been
considered (Gumbel, 1954a; 1958). Each of these types results in a different functional
form of the extreme value asymptotes. The first type, which is called the exponential
type , comprises those distributions that for large x converge to unity at least as fast as the
exponential function itself; all their moments exist. These types of distribution satisfy
→∞
f ( x )
f ( x )
f ( x )
f ( x )
f ( x )
F ( x ) =
f ( x )
F ( x ) =
and
(13.45)
1
for very large and for very small values of x , respectively. Since both numerator and
denominator in these ratios are very small, this suggests the application of de L'Hospital's
rule; hence one can continue the process, to obtain for very large x ,
f ( x )
f ( x )
f ( x )
f ( x )
f ( x )
F ( x ) =
=
=···
(13.46)
1
and so on, and an analogous result for very small x . Examples of this type of distri-
bution are the normal, the logistic, the gamma, and their logarithmically transformed
distributions.
The distributions of the second type are also referred to as Cauchy type distributions;
these are distributions, which do not have moments above a certain order. The lim-
ited distributions belong to the third type of initial distributions; these are distributions
with an upper or lower bound or with both. In hydrology they are mainly of interest in
the analysis of low flows and droughts. This brings up the point that the classification
into these three separate types of initial distributions is not always rigid. For exam-
ple, the lognormal distribution is of the exponential type at the upper end, since x can
assume values all the way to infinity; however, it is of the limited type at the lower
end of the distribution, because x cannot be smaller than zero or c , as can be seen in
Equations (13.38) and (13.39).
The first asymptote for largest values
This distribution is also often referred to as the Gumbel distribution after the statistician
who clarified and promoted its use (Gumbel, 1954a; 1958). Several derivations have
been presented in the literature. One of the simpler derivations proceeds as follows. The
starting point is the Taylor expansion of the initial distribution about some characteristic
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