Geoscience Reference
In-Depth Information
4.1.3 Bayesian Test
The Bayesian test was developed by Chernoff and Zacks (1964), which was
modified later by Gardner (1969). The Gardner's test-statistic ( G ) for a two-
sided test on a shift in the mean at an unknown point can be written as
(Gardner, 1969):
n
1
2
Ç
*
kkY
G =
pS
V
(7)
k
1
where p k = prior probability that the shift occurs just after k th observation.
Here, it is assumed that the population variance
()
Y V is known. If
V is not
known, it can be replaced with the sample variance. For p k independent of k ,
the test-statistic, U can be expressed as:
n
1
1
2
Ç
* k
S
U =
(8)
nn
(
)
k
1
However, for p k proportional to [ k ( n - k )] -1 , the test-statistic can be written
as:
n
1
2
Ç
**
k
Z
,
A =
k = 1, 2, ......, n
(9)
k
1
where
Z = weighted rescaled partial sums, which can be computed using the
following formula:
**
` 12
Ë
^
k
Û
kn k
S
D
**
k
Z
=
(10)
Í
Ý
x
Large values of U and A test-statistics indicate departures from the
homogeneity, which is judged based on their critical values (Buishand, 1982).
Buishand (1982) reported that the tests based on the cumulative deviations
are superior to the von Neumann test for a model with only one change in the
mean. The tests were applied to the annual rainfall data of 264 rainfall stations
in the Netherlands, and departures from homogeneity were found. The von
Neumann test provided almost the same results as the tests based on the
cumulative deviations.
4.1.4 Tukey Test for Multiple Comparisons
This test is used to examine the significance of all possible differences among
different population means. The size of the different samples may be unequal
but all populations should be normally distributed with equal variances. Hence,
it is a parametric test, which depends upon the distribution parameters. To
apply this test on a hydrologic time series x t ( t = 1, 2, …, n ), the entire series
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