(1) The order statistic approach relies on a very

This expression gives the probability that the

nonexceedance probability associated with *Y*p 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 ( Y*j ).

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

(4)

ordered as *Y*j , *j *= 1,2,3,...n, *Y*j # *Y*j + 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

associated with the jth

observed data point

' *P *[ j or more observations $ *Y*p ]

associated with *Y*p

where *Y*p = 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 [F*e ( Yp ) < Fe ( Yj )],

or confidence level, is associated with a rank *j*, and

consequently a ranked observation *Y*j . Therefore,

[ *F *( *Y*p) ]i [ 1 & *F *( *Y*p) ]n&*i*

(1)

estimates of uncertainty can only be provided corre-

sponding to the range of the *Y*j . This is shown in

Figure 1. Note that the uncertainty distribution cannot

where *p = F ( Y*p ) = 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 *Y*p , 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

m

1

(2)

in the estimated frequency curve over this range of

probabilities.

' *I B*p( *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

2

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