Environmental Engineering Reference
In-Depth Information
where
ˆ
µ
x
and
ˆ
σ
x
are the maximum likelihood esti-
mates of
μ
x
and
σ
x
, respectively. Solving Equations
(10.66) and (10.67) simultaneously yields
λ
=
6
β
−
6
β β
+
(10.71)
3
2
1
0
λ
=
20
β
−
30
β
+
12
β β
−
(10.72)
4
3
2
1
0
N
1
Since the probability-weighted moments involve
raising values of
F
X
rather than
x
to powers, and because
F
X
≤ 1, then estimates of probability-weighted moments
and l-moments are much less susceptible to the influ-
ences of a few large or small values in the sample. Hence,
l-moments are generally preferable to product moments
for estimating the parameters of probability distribu-
tions of water-quality variables.
Consider a sample of
N
measured values of a random
variable
X
. To estimate the l-moments of the probabil-
ity distribution from which the sample was taken, first
rank the values as
x
1
≤
x
2
≤
x
3
≤ ··· ≤
x
N
, and estimate the
probability-weighted moments as follows (Hosking and
Wallis, 1997):
∑
ˆ
µ
x
=
x
i
N
i
=
1
N
1
∑
(
)
2
ˆ
ˆ
σ
=
x
−
µ
x
i
x
N
i
=
1
It is apparent from these results that the maximum-
likelihood estimate of
μ
x
is the same as that using the
method of moments, while the maximum likelihood
estimate of
σ
x
is different from the method-of-moments
estimate by a factor of
N
/ − 1
. These results are
applicable only for samples drawn from a normal
distribution.
N
10.6.3 Method of L-Moments
1
∑
b
=
x
i
(10.73)
0
N
The typically small sample sizes available for character-
izing water-quality variables yield estimates of the third
and higher moments that are usually very uncertain.
This has led to the use of an alternative system for esti-
mating the parameters of probability distributions
called
L-moments
. The
r
th probability-weighted
moment,
β
r
, is defined by the relation
i
=
1
N
1
∑
b
=
(
i
−
1
)
x
i
(10.74)
1
N N
(
−
1
)
i
=
2
N
1
1
∑
b
=
(
i
−
1
)(
i
−
2
)
x
i
(10.75)
2
N N
(
−
)(
N
−
2
)
i
=
3
N
1
+∞
∫
∑
β
r
=
x F x
[
( )]
r
f
( )
x dx
(10.68)
b
=
(
i
−
1
)(
i
−
2
)(
i
−
3
)
x
i
X
X
3
N N
(
−
1
)(
N
−
2
)(
N
−
3
)
−∞
i
=
4
(10.76)
where
F
X
and
f
X
are the CDF and probability density
function of
x
, respectively. The l-moments,
λ
r
, are linear
combinations of the probability-weighted moments,
β
r
,
and the first four l-moments are computed as
The sample estimates of the first four l-moments,
denoted by
L
1
to
L
4
, are then calculated by substituting
b
0
to
b
3
for
β
0
to
β
3
, respectively, in Equations (10.69-
10.72). The l-moments of the normal and log-normal
probability distributions are given in terms of the
parameters of the distributions in Table 10.5. Equating
λ
=
β
(10.69)
1
0
λ
=
2
β β
−
(10.70)
2
1
0
TABLE 10.5. Moments and L-Moments of Common Probability Distributions
Distribution
Parameters
Moments
l-Moments
normal
μ
X
,
σ
X
μ
X
=
μ
X
λ
1
=
μ
X
σ
π
X
σ
X
=
σ
X
λ
2
=
σ
2
Y
log-normal (
Y
= ln
X
)
μ
Y
,
σ
Y
μ
Y
=
μ
Y
λ
=
exp
µ
+
1
Y
2
σ
2
σ
σ
Y
=
σ
Y
Y
Y
λ
=
exp
µ
+
erf
2
Y
2
2
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