FLOOD00764

• County: Statewide
• Community: Nationwide
• Title: Use of Extreme Value Theory in Estimating Flood Peaks from Mixed Populations
• Date: 2/1/1980
• Prepared By: Utah State University Water Research Laboratory
• Floodplain - Doc Type: Educational/Technical/Reference Information

EMPIRICAL PIT

The problem encountered when emp ir ieal
fit 15 the sole criterion used to select a
"best" distribution to describe a population
increases as one uses the distribution to
estimate the frequency of rarer events. It
is sometimes suggested that no distri-
bution is perfect; therefore, several may do
an adequate job, and certainly the "best" fit
will be close. This argument may be valid
when the distributions are used to estimate
probabilities or return periods for frequent-
ly occurring events. However, when estimates
are needed for extreme or rare events,
serious errors can result from use of a
distribution selected on the basis of empiri-
cal fit because the probabilities of rare
events are computed from the tails of a
distribution, whereas empirical fit IS
dominated by the body of the data set. The
following Monte Carlo experiment was perform-
ed to provide some idea of the magnitude of
the problem.

Twenty
25 values,
population
function

random samples, each containing
were generated from a Weibull
with cumulative distribution

F(x)

{ : - exp[-(x/30)b]

x , 0

x ;:.: 0

The ~amma distribution is considered close to
the Weibull (Hager, Bain, and Antle 1971) and
IS a likely alternative for fitting such
data. Both gamma and Weibull distributions
were fit to the data sets. The method of
White (1969) was used to estimate Weibull
parameters, and the method of moments (Lind-
gren 1976) was used for the gamma distribu-
t ion. Let FW(x) and FG(x} denote the We ibull
and gamma distribution functions respectively
with parameter values estimated from data.

Goodness of f it is based upon the
empirical distribution

FS(x)

{O'
= :/n

X(n)

< x

xO)
x(i) $ x < x(i+1)

i = 1,2, ..., n

(1)

where x(I), x(2),
data values. Two
to judge the fit.
t ions, i.e.,

..., x (n) are the orde-red
common crIteria were used
The sum of squared devia-

ss

2
E(FW(x(i)) - FS(x(i)))

for the Weibull fit or

2
SS = E(FG(x(i)) - FS(x(i)))

for the ~amma fit. The second measure is a
Kolomogorov type (denoted K) where

K E1Fw(x(i)) - FS(x(i)) I

or

K EIFG(X(i)) - FS(x(i)) I

for the Weibull or gamma distributions
res pect i ve ly.

AccordIng to the first measure of fit
(55), three tImes out of the 20 runs the
gamma exhibited the better fit. In eight out
of the 20 runs, the second measure (K) showed
the gamma as having the better fit. This
frequency of misclassification demonstrates a
real possibility of selecting the wron~
distribution with real data.

The log-Pearson type III distribution
is the most widely used for flood frequency
analysis. It has been chosen from among
several candidate distributions by first
estimating the parameters of each distribu-
t ion for each of a large number of gaged
records (Benson 1968). Then a goodness-of-fit
criterion which emphasizes selected flood
flows from 2 to 100 years (U.S. Water Re-
sources Council 1976, Appendix 14) was used
to select the best overall fit. Although
selection of the log-Pearson type III is
based upon f it in the right tail, est imat ion
of parameters for each distribution IS by
standard methods which emphasizes fit in the
body of the data. In certain cases, the fit
in the right tail is poor. Even if the fit
is good, blind application of a distribution
selected on the basis of empirIcal fit can
lead to serious error. The magnitude of this
error 18 illustrated in the following ex-
ample. The 99th percentile was computed from
both the Weibull and gamma estimated distri-
bution for each of the 20 data sets. The
results are summarized in Table 1. In every
case the gamma distributed percentile
exceeded the true value and the Weibull
estimated value. The average Weibull est i-
mate also exceeds the true value, however the
amount is within the expected sampling

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