Environmental Engineering Reference
In-Depth Information
perform poorly in estimating the frequencies of out-
comes in the tails of the fitted distributions (e.g., Mitosek
et al., 2006). In cases where there are several competing
distributions, goodness-of-fit statistics can be used to
rank the effectiveness of the competing distributions.
10.6 ESTIMATION OF PARAMETERS OF
POPULATION DISTRIBUTION
The population probability distribution of a random
variable
X
is typically written in the form
f
X
(
x
). However,
this distribution could more explicitly be written in the
form
f
X
(
x
|
θ
1
,
θ
2
, . . . ,
θ
m
) to indicate that the probability
distribution of the random variable also depends on the
values of the
m
parameters
θ
1
,
θ
2
, . . . ,
θ
m
. This explicit
expression for the probability distribution is particularly
relevant when the population parameters are estimated
using sample statistics, which are themselves random
variables. The most common methods for estimating the
parameters of probability distributions from measured
data are: the method of moments, the maximum likeli-
hood method, and the method of l-moments.
EXAMPLE 10.9
The set of concentration measurements (in mg/l) shown
in Table 10.3 were derived from samples over a 50-
month period.
use the KS statistic to assess the hypothesis that
these samples are drawn from a log-normal distribution
with a (natural-log) mean of 1.2 and a standard devia-
tion of 0.8.
10.6.1 Method of Moments
Solution
The
method of moments
is based on the observation that
the parameters of a probability distribution can usually
be expressed in terms of the first few moments of the
distribution. These moments can be estimated using
sample statistics, and then the parameters of the distri-
bution can be calculated using the relationship between
the population parameters and the moments. The three
moments most often used are the mean, standard devia-
tion, and skewness, defined by Equations (10.10), (10.12),
and (10.14) for continuous probability distributions. In
practical applications, these moments are estimated
from finite samples, and the following equations usually
provide the best estimates:
From the given data:
μ
y
= 1.2,
σ
y
= 0.8, and
N
= 50.
ranking the concentration data,
c
, lowest to highest,
calculating the corresponding log-normal cumulative
distribution,
P
X
(
c
), and calculating the sample cumula-
tive distribution,
S
N
(
c
) (=
k
/
N
), yields the results shown
in Table 10.4.
Based on these results, the maximum value of
|
P
X
−
S
N
| is 0.061. According to the critical KS values
given in Appendix C.5, for a 5% level of significance,
the KS statistic is 0.190. Since 0.061 < 1.190, the
hypothesis that the data is log-normally distributed is
accepted.
TABLE 10.3. Measurements of Concentration
2.44
10.44
2.68
6.18
17.62
15.64
8.33
1.42
2.66
2.90
3.51
18.79
6.67
0.89
3.52
2.53
7.27
7.80
3.35
3.62
2.19
0.78
3.54
8.55
5.45
5.00
1.79
3.75
1.53
1.24
2.12
3.25
7.66
6.89
1.22
1.09
2.19
6.65
4.62
4.87
1.02
0.71
4.46
3.67
1.70
1.74
1.97
6.82
6.34
2.43
TABLE 10.4. Statistical Properties of Concentration Measurements
c
(mg/l)
c
(mg/l)
k
P
X
(
c
)
S
N
(
c
)
|
P
X
−
S
N
|
k
P
X
(
c
)
S
N
(
c
)
|
P
X
−
S
N
|
1
0.71
0.027
0.020
0.007
26
3.52
0.529
0.520
0.009
2
0.78
0.035
0.040
0.005
27
3.54
0.533
0.540
0.007
3
0.89
0.051
0.060
0.009
28
3.62
0.542
0.560
0.018
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
0.061
11
1.74
0.209
0.220
0.011
36
6.18
0.781
0.720
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
23
3.25
0.490
0.460
0.030
48
15.64
0.974
0.960
0.014
24
3.35
0.504
0.480
0.024
49
17.62
0.982
0.980
0.002
25
3.51
0.528
0.500
0.028
50
18.79
0.985
1.000
0.015
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