Geoscience Reference
In-Depth Information
is applicable to describe maxima when their initial distribution has an upper bound; if
this upper bound is denoted by
ω
, it follows that the initial distribution is subject to
F
(
x
)
=
1
for
x
=
ω
Several different derivations have been presented in the literature for this asymptote
(see Gumbel, 1958, p. 273 ff.), which require certain assumptions regarding the manner
in which
F
(
x
) approaches unity. Probably the simplest, after Kimball (1942), consists
of observing that the bounded variable
x
≤
ω
can be transformed into an unbounded
variable
z
as follows
ω
−
x
=
a
exp[
−
b
(
z
−
c
)]
(13.61)
,
→∞
→
ω
where
a
. If it can be assumed
that the resulting distribution function of
z
is of the exponential type, then according to
(13.49), the initial distribution of
x
in the neighborhood of the upper bound
x
b
and
c
are constants; this shows how
z
as
x
=
ω
, can
be described by
ω
−
k
1
n
x
ω
−
v
F
(
x
)
=
1
−
(13.62)
where the parameters in (13.61) have been changed to
a
u
n
.
Proceeding as before, one finds immediately that the probability that all items in a very
large sample are smaller than or equal to
x
, in the limit as
n
=
ω
−
v,
b
=
α
n
/
k
and
c
=
→∞
,is
exp
k
ω
−
x
ω
−
v
=
−
G
3
(
x
)
(13.63)
G
3
(
x
)is
The corresponding third asymptotic density function
g
3
(
x
)
=
ω
−
k
−
1
k
ω
−
v
x
ω
−
v
g
3
(
x
)
=
G
3
(
x
)
(13.64)
The moments of the third asymptote are treated in detail below for the smallest
values.
The third asymptote for smallest values
In hydrology it is mainly the third asymptotic distribution for smallest values that has
been of interest. Indeed while, in principle at least, different types of common events,
such as rainfall amounts, wind speeds or river flows, can often be assumed to be unlimited
in magnitude, even the smallest of such events can never be smaller than zero. Thus the
smallest values often have a definite lower limit below which they cannot go. As for the
first asymptote, the symmetry principle (13.60) can be applied to derive the distribution of
the smallest values from that of the largest values. The procedure consists of changing the
sign of
x
,ω
and
v
, and then assigning different values to the parameters, say
ω
1
and
v
1
,
to obtain
exp
k
x
−
ω
1
v
1
−
ω
1
=
−
−
1
G
3
(
x
)
1
(13.65)
