ETL 1110-2-537
31 Oct 97
b. Order statistics.
IB p ( j, n - j + 1 ) = incomplete beta function
(1) The order statistic approach relies on a very
This expression gives the probability that the
nonexceedance probability associated with Yp is
straightforward application of the binomial
distribution (Mood, Graybill, and Boes 1963, p 513).
greater than or equal to the nonexceedance probability
associated with the jth ordered observation F ( Yj ).
The problem is to calculate the probability that the
flow or stage corresponding to an exceedance
The equivalent relationship for exceedance
probability exceeds a particular value. The only values
probabilities is
available are the observations of the random variable.
Assume that the observations of flow or stage are
P [ Fe ( Yp ) # Fe ( Yj ) ] ' I Bp ( j, n & j % 1 )
(4)
ordered as Yj , j = 1,2,3,...n, Yj # Yj + 1 and n is the
number of years of record. The uncertainty about the
frequency curve at nonexceedance probability p is
where
estimated by the probability
Fe ( Yj ) = exceedance probability
P [ Yp $ Yj ] ' P [ jth order statistic # Yp ]
associated with the jth
observed data point
' P [ j or more observations $ Yp ]
Fe ( Yp ) = ( 1 - p ) = exceedance probability
associated with Yp
where Yp = quantile (e.g., flow or stage) for
nonexceedance probability p. Applying the binomial
(3) The important characteristic of these equations
theorem
is that the distribution estimate P [Fe ( Yp ) < Fe ( Yj )],
or confidence level, is associated with a rank j, and
consequently a ranked observation Yj . Therefore,
n
n
[ F ( Yp) ]i [ 1 & F ( Yp) ]n&i
(1)
P [ Yp $ Yj ] ' E
estimates of uncertainty can only be provided corre-
i
i'j
sponding to the range of the Yj . This is shown in
Figure 1. Note that the uncertainty distribution cannot
where p = F ( Yp ) = nonexceedance probability
be extrapolated beyond the smallest or largest
associated with the quantile of interest. The
observation. If only 20 observations exist, then the
nonexceedance probability or a corresponding
maximum value for any uncertainty distribution level
exceedance probability can be calculated by use of a
is the largest out of 20 observations.
plotting position formula.
(4) Despite this limitation, the order statistic
(2) This computation is rather inconvenient,
estimate is useful because it is sensitive to local
although it can be performed fairly easily by computer.
changes of slope in the frequency curve. This means
A more convenient and equivalent expression which
the method is able to account for the local variability
estimates the uncertainty in the estimate of the
of a random variable. For example, a regulated
nonexceedance probability p for quantile Yp , involves
frequency curve may have very small slope over a
the incomplete beta function (Press et al. 1990, p 166)
significant range in probability indicating very little
variability in releases. The order statistic approach
will recognize this and predict very little uncertainty
p
m
1
z j&1 (1 & z)n& jd z
P [ F (Yp) $ F ( Yj) ] '
(2)
in the estimated frequency curve over this range of
B ( j, n & j % 1 ) 0
probabilities.
c. Example. As a simple example of the
' I Bp( j, n & j % 1 )
(3)
application of Equation 1, assume that there are 5
years of observed peak annual stages. The estimated
nonexceedance probability corresponding to the largest
where
observed event is computed by the Weibull plotting
position as
F (Yp ) = p B ( j, n - j + 1 ) = beta function
2
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