Geoscience Reference
In-Depth Information
costly natural disaster in the United States. Tropical storms or hurricanes rarely pene-
trate this far north in full strength, but when they do they can materially affect the annual
mean.
13.4.2
The lognormal distribution
Many natural phenomena, which have a lower bound and exhibit positive skew, cannot
be described well by the normal distribution, but in some cases their logarithms can.
According to the Central Limit Theorem, this would be the case when the random
variable is the product of
n
variables, each with its own arbitrary density function with a
finite mean and variance. Upon applying Equation (13.24) with
y
=
ln
z
to (13.34), the
lognormal density function results, and it can be written as
exp
2
ln(
x
)
1
σ
n
x
√
2
1
2
−
μ
n
σ
n
f
(
x
)
=
−
−∞
<
ln(
x
)
≤∞
(13.38)
π
where
σ
n
are the mean and the standard deviation of ln(
x
). When the logarithms
of the data do not quite follow the normal distribution, introduction of a lower bound
c
,
different from zero, may improve the fit. The density function becomes in this case
μ
n
and
exp
2
y
n
−
μ
nc
σ
nc
1
1
2
f
(
x
)
=
c
)
√
2
−
−∞
<
y
n
≤∞
(13.39)
σ
nc
(
x
−
π
in which
y
n
=
σ
nc
are the mean and standard deviation of
y
n
.
When
c
is known (e.g. from physical considerations about the lower bound), these two
parameters can be estimatated by means of the first and second of (13.13), after replacing
X
i
by ln(
X
i
−
ln(
x
−
c
), and
μ
nc
and
c
); alternatively, they can also be estimated directly from
μ
and
σ
,by
inversion of the following two equations (see, for example, Chow, 1954)
exp
μ
nc
+
nc
μ
=
c
+
0
.
5
σ
(13.40)
c
)
2
exp(
1
2
nc
)
σ
=
(
μ
−
σ
−
However, because the moments of the
X
i
values are different from those of the ln(
X
i
−
c
)
values, these two procedures yield different results; Stedinger (1980) has shown that for
smaller samples the procedure with the moments of the logs yields better parameter
estimates than that based on Equations (13.40).
The determination of
c
is not always easy. A rough idea of its value can be obtained
graphically by plotting the data on log-normal probability paper, i.e. ln(
X
i
−
c
) versus
F
Y
(
y
) (see Equation (13.28)) for different trial values of
c
; that value of
c
is selected which
produces the best straight line. The value of
c
can in principle be obtained by the method
of moments. Since the first two moments are already used in (13.40) to determine
μ
and
σ
,
the third moment is needed. In the case of the lognormal distribution (see, for example,
Chow, 1954), this is related to the second as follows
C
s
=
3
C
v
+
C
v
(13.41)
