Part C | 26.3
578 Part C Coastal Design
The fundamental concept for the GEV is based on
the analysis of asymptotic characteristics of probability functions. In this context, two CDFs are said to be
of the same type if there exists constants a > 0 and b
such that F 1 .ax C b/ D F 2 .x/, where the subscripts denote the different CDFs. It was shown by Fisher and
Tippett [26.45] that, excluding certain improper distributions characterized by discontinuities in the CDF,
there are only three possible limiting distribution types
which meet this constraint. Jenkinson [26.49] showed
that all three of these asymptotic distributions could be
written in a common form
F.x/ D exp
"
1 C "
 .x c/
d
à 1
"
#
;
(26.7)
where c, d, and " are parameters of the distribution.
For the case of " D 0, this equation is defined over
a range of x from 1 to C1 and reduces to the twoparameter Gumbel distribution, sometimes termed the
Fisher–Tippett Type I distribution.
F.x/ D expΠexp.y/; where y D
x c
d
:
(26.8)
For " > 0, the resulting distribution is a Frechet (or
Fisher–Tippett Type II) distribution, which is bounded
at the low end of the distribution but is unbounded
for large x; and for " < 0, the resulting distribution is
a Weibull (or Fisher–Tippett Type III) distribution. The
Weibull distribution has an explicit upper limit, but is
unbounded in the negative direction.
Returning to Figs. 26.1 and 26.2, we see why a simple plot of x versus lnfflnŒF.x/g will form a straight
line if the sample follows a Gumbel distribution. If the
points tend to curve upward on such a plot, the distribution is of the Fisher–Tippett Type III form; and if the
points tend to curve downward, the distribution is of the
Fisher–Tippet Type II form, which predicts larger values of x than the Gumbel distribution for a given value
of the CDF.
Shortly after applications with GEV distributions
began to become widespread, a second distribution
with a somewhat different theoretical basis was introduced. This distribution, the generalized Pareto distribution (GPD), was first published by Pickands [26.50]
and has been further developed by many subsequent
researchers, with some of the more topical for our
purposes being Resnick [26.14] and Davison and
Smith [26.51]. Methods for fitting the GPD can be
found in articles such as [26.52] and are now available
in many statistical packages. The GPD can be written
in the form
F.xjx c / D 1
Â
1 C "
0 x c
0
à 1
" 0
;
(26.9)
which has three parameters ("
0 ,
0 , and x c ), similar to
the GEV, and also similar to GEV; the GPD has three
basic forms depending on ". With "
0
D 0, the GPD approaches an exponential form given by
F.xjx c / D 1 exp
Â
x
0
Ã
:
(26.10)
Since the exponential and double exponential (Gumbel)
distributions have been shown to converge for mean recurrence intervals greater than 7 years (based on annual
data), significant deviations between the GPD and GEV
for the case of " D "
0
D 0 tend toward zero for estimates
of values with mean recurrence intervals greater than 7.
When "
0 is positive, the GPD will be similar to a Fisher–
Tippett Type II and when "
0 is negative, the GPD will
be similar to a Fisher–Tippett Type III.
It is sometimes argued that the GPD is superior to
the GEV, due to its formal incorporation of a threshold into its derivation; however, the essence of this
distinction lies in the assumption that a sample of values from nature represents a homogeneous population
in a statistical sense. As will be shown later in this
chapter in Sect. 26.3.4, samples in nature often contain a mixture of populations. For example, large surges
and waves can be generated along coasts by both extratropical and tropical storms. In smaller basins, squall
lines and thunderstorms can become dominant wave
and surge producers. Wherever possible it is advisable
to try to avoid mixing populations from clearly different sources, but sometimes the ability to recognize more
subtle variations can be obscured. A good example of
this can be found in the work of Resio [26.53], who
showed that what appeared to be a Fisher–Tippett II
distribution for hindcast wave heights in the Cape Hatteras area was actually the sum of different types of
storms definable in terms of how far off the coast they
were when they passed Cape Hatteras. Estimated values
within the record length are usually not affected by this
mixed population problem however, it can become very
significant when extrapolating to larger values.
Given the previous discussion, and hopefully without unduly disparaging the excellent theoretical work
which has been accomplished in the field of extreme
statistics, it should be recognized that given the nuances in the different theories upon which the GEV
and GPD are based, it is difficult to argue definitively
that one class of distribution is superior in applications
to the other. Both distributions represent either a twoparameter distribution (for the limiting case " D "
0
D 0)
578 Part C Coastal Design
The fundamental concept for the GEV is based on
the analysis of asymptotic characteristics of probability functions. In this context, two CDFs are said to be
of the same type if there exists constants a > 0 and b
such that F 1 .ax C b/ D F 2 .x/, where the subscripts denote the different CDFs. It was shown by Fisher and
Tippett [26.45] that, excluding certain improper distributions characterized by discontinuities in the CDF,
there are only three possible limiting distribution types
which meet this constraint. Jenkinson [26.49] showed
that all three of these asymptotic distributions could be
written in a common form
F.x/ D exp
"
1 C "
 .x c/
d
à 1
"
#
;
(26.7)
where c, d, and " are parameters of the distribution.
For the case of " D 0, this equation is defined over
a range of x from 1 to C1 and reduces to the twoparameter Gumbel distribution, sometimes termed the
Fisher–Tippett Type I distribution.
F.x/ D expΠexp.y/; where y D
x c
d
:
(26.8)
For " > 0, the resulting distribution is a Frechet (or
Fisher–Tippett Type II) distribution, which is bounded
at the low end of the distribution but is unbounded
for large x; and for " < 0, the resulting distribution is
a Weibull (or Fisher–Tippett Type III) distribution. The
Weibull distribution has an explicit upper limit, but is
unbounded in the negative direction.
Returning to Figs. 26.1 and 26.2, we see why a simple plot of x versus lnfflnŒF.x/g will form a straight
line if the sample follows a Gumbel distribution. If the
points tend to curve upward on such a plot, the distribution is of the Fisher–Tippett Type III form; and if the
points tend to curve downward, the distribution is of the
Fisher–Tippet Type II form, which predicts larger values of x than the Gumbel distribution for a given value
of the CDF.
Shortly after applications with GEV distributions
began to become widespread, a second distribution
with a somewhat different theoretical basis was introduced. This distribution, the generalized Pareto distribution (GPD), was first published by Pickands [26.50]
and has been further developed by many subsequent
researchers, with some of the more topical for our
purposes being Resnick [26.14] and Davison and
Smith [26.51]. Methods for fitting the GPD can be
found in articles such as [26.52] and are now available
in many statistical packages. The GPD can be written
in the form
F.xjx c / D 1
Â
1 C "
0 x c
0
à 1
" 0
;
(26.9)
which has three parameters ("
0 ,
0 , and x c ), similar to
the GEV, and also similar to GEV; the GPD has three
basic forms depending on ". With "
0
D 0, the GPD approaches an exponential form given by
F.xjx c / D 1 exp
Â
x
0
Ã
:
(26.10)
Since the exponential and double exponential (Gumbel)
distributions have been shown to converge for mean recurrence intervals greater than 7 years (based on annual
data), significant deviations between the GPD and GEV
for the case of " D "
0
D 0 tend toward zero for estimates
of values with mean recurrence intervals greater than 7.
When "
0 is positive, the GPD will be similar to a Fisher–
Tippett Type II and when "
0 is negative, the GPD will
be similar to a Fisher–Tippett Type III.
It is sometimes argued that the GPD is superior to
the GEV, due to its formal incorporation of a threshold into its derivation; however, the essence of this
distinction lies in the assumption that a sample of values from nature represents a homogeneous population
in a statistical sense. As will be shown later in this
chapter in Sect. 26.3.4, samples in nature often contain a mixture of populations. For example, large surges
and waves can be generated along coasts by both extratropical and tropical storms. In smaller basins, squall
lines and thunderstorms can become dominant wave
and surge producers. Wherever possible it is advisable
to try to avoid mixing populations from clearly different sources, but sometimes the ability to recognize more
subtle variations can be obscured. A good example of
this can be found in the work of Resio [26.53], who
showed that what appeared to be a Fisher–Tippett II
distribution for hindcast wave heights in the Cape Hatteras area was actually the sum of different types of
storms definable in terms of how far off the coast they
were when they passed Cape Hatteras. Estimated values
within the record length are usually not affected by this
mixed population problem however, it can become very
significant when extrapolating to larger values.
Given the previous discussion, and hopefully without unduly disparaging the excellent theoretical work
which has been accomplished in the field of extreme
statistics, it should be recognized that given the nuances in the different theories upon which the GEV
and GPD are based, it is difficult to argue definitively
that one class of distribution is superior in applications
to the other. Both distributions represent either a twoparameter distribution (for the limiting case " D "
0
D 0)