Geoscience Reference
In-Depth Information
Fig. 13.11 The parameter
a
in the
generalized extreme
value distribution
(13.74), as a function of
the skew coefficient
C
s
.
0.3
0.2
a
0.1
0
−
0.1
−
0.2
−
0.3
0
5
10
15
C
s
or
∞
m
n
=
x
1
)
n
exp[
b
)
1
/
a
(
x
x
1
)
1
/
a
]
d
[
b
)
1
/
a
(
x
x
1
)
1
/
a
]
−
−
−
/
−
−
−
/
−
(
x
(
a
(
a
x
1
(13.77)
This produces finally
m
n
=
a
)
n
(
−
b
/
(1
+
an
)
with
a
>
−
1
/
n
(13.78)
Hence the mean is
m
1
+
x
1
,or
μ
=
+
/
a
)
(
1
−
+
a
)
)
c
(
b
(1
(13.79)
Similarly, on account of (13.12) the variance is
2
a
)
2
[
a
))
2
]
σ
=
(
b
/
(1
+
2
a
)
−
(
(1
+
(13.80)
and the third central moment is
a
)
3
[
a
))
3
]
m
3
=−
(
b
/
(1
+
3
a
)
−
3
(1
+
a
)
(1
+
2
a
)
+
2 (
(1
+
(13.81)
b
and
c
.
First, the parameter
a
can be determined by iteration from the sample skew coefficient
g
s
(see Equation (13.13)), expressed in terms of the ratio of Equations (13.81) and (13.80);
a rough idea of the magnitude of
a
can be obtained from Figure 13.11. With this result
b
can be obtained from the sample variance
S
2
and (13.80), and then
c
from the sam-
ple mean
M
with (13.79). If the data record is so short that the third moment must be
considered unreliable, one can also apply the Weibull procedure, explained earlier for
the third asymptote for smallest values. In brief, this consists of observing that Equa-
tion (13.74) produces
F
(
c
)
As before, these first three moments can be used to estimate the parameters
a
,
1); thus the parameter
c
can be estimated immedi-
ately from the available data as the value of
x
, which corresponds with a probability
=
exp(
−
