Geoscience Reference
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For 5 n 40, critical values of the test-statistic can be obtained from
Kanji (2001). For n > 40, ' K ' may be assumed to follow a normal distribution
with the mean (P k ) and variance (V 2 k ) given as:
21
3
n
P k =
(53)
16
n
29
V 2 k =
(54)
90
Now, the test-statistic ( z ) for the case when n > 40 is computed by the
following expression:
K P
V
0.5
k
k
z =
(55)
Critical values of ' z ' can be obtained from the standard normal distribution
tables available in statistics topics. For both the cases of time series sizes, if
the computed test-statistic values are less than its critical values, the null
hypothesis cannot be rejected. That is, the time series is considered to be
random.
4.3.10 Wilcoxon-Mann-Whitney Rank Sum Test
This is a nonparametric test (i.e., distribution-free), which is applicable if the
observations are random and independent. It is used to examine whether the
occurrence of increasing or decreasing successive values of a time series is
random. Consider a time series x t ( t = 1, 2, …, n ). First, successive observations
in the sequence are coded with a '+' or '-' sign by comparing two successive
values, and the ranks (1, 2, 3, …, n ) are assigned to all the observations of the
series. Thereafter, the number of '+' and '-' signs are counted and the larger
of two numbers is noted down (say n 1 ). If n 2 be the number of opposite signs,
then n = n 1 + n 2 . Now, from the integers describing the natural order of signs,
the rank sum ' R 1 ' of ' n 2 ' signs is determined. Finally, the value of ' R 2 ' statistic
is calculated by the following expression:
R 2 =
nn
1
R
(56)
2
1
The smaller of ' R 1 ' and ' R 2 ' is used as the test-statistic. The critical values
of the test-statistic can be found in Natrella (1963). If the computed value of
the test-statistic is greater than its critical value, the null hypothesis of no
trend is rejected.
4.3.11 Inversions Test
This test is used to examine the linear trend in a time series. It is almost
similar to the Kendall's rank correlation test (described ahead). The number of
x j -values ( j > i ) each smaller than a chosen x i -value is counted for all i 's
 
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