Environmental Engineering Reference
In-Depth Information
For the standard deviation σ, a percentile method is proposed by Donoho and Johnstone
(1994): σ can be estimated as the sample median of the absolute values of the deviations
from the sample mean m, divided by 0.6745. Namely,
Samplemedian of Y
()
1
m,Ym,
()
2
, Ym
(k)
(1.14)
σ≈
0 6745
.
For normal distribution, the sample mean in Equation 1.14 can also be estimated as the
sample median. Table 1.3 shows the |Y-m| data and the sorted data ( m = 97.97 is the sample
median of Y). The sample median of the sorted |Y-m| data is (16.82 + 23.57)/2 = 20.19. The
percentile estimate for σ is therefore 20.19/0.6745 = 29.94.
The percentile method of estimating μ and σ is more robust than the method of moments,
as outliers will not affect the point estimates significantly. For example, if the maximum
value is recorded wrongly by a factor of 10, the sample median value remains the same
(=97.97), but the sample mean ( m ) is now 223.12. The actual mean is 100. The estimate
from Equation 1.14 still remains the same (=29.94), but the sample standard deviation ( s )
is now 295.61. Robust statistics is an important practical topic. The reader can refer to
advanced texts for more details.
1.2.3.3 Maximum likelihood method
The principle of maximum likelihood says that the best estimates for μ and σ are those that
maximize the likelihood function. For independent observations (Y (1) , …, Y ( n ) ), the likeli-
hood function is the joint density of (Y (1) , …, Y ( n ) )
n
n
1
−−
(
Y
()
k
µ
)
2
()
1
()
2
()
n
()
k
f
(
YY
,
,
,
Y
|
µσ
,)
=
f
(
Y
|
µσ
, )
=
exp
(1.15)
2 ⋅
σ
2
2
πσ
k
=
1
k
=
1
Equivalently, one can find the maximum likelihood estimates for μ and σ by maximizing
the logarithm of the likelihood function
n
(
Y
()
k
µ
)
2
k
()
1
()
2
()
n
ln[(
f
YY
,
,
,
Y|
µσ
, )]
=
05
. n(
2
π
)ln( )
σ
(1.16)
2
σ
2
=
1
Table 1.3 Sample values of |Y- m | ( m = sample median = 97.97)
|Y (k) -m|
Index k
Sorted |Y-m| data
Rank
26.10
1
7.45
1
15.39
2
7.45
2
36.78
3
8.15
3
7.45
4
15.39
4
35.58
5
16.82
5
23.57
6
23.57
6
16.82
7
26.10
7
7.45
8
26.54
8
26.54
9
35.58
9
8.15
10
36.78
10
 
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