Statistical Analysis of Trends in the Red River Over a 45 Year Period

variance of S is computed;
Var(S) =
n(n − 1)(2n + 5)
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and the test statistic Z, is as follows;
When there are tied values,
Var(S) = 1 18
[
⎧
S − 1
, if S > 0;
⎪⎨ [Var(S)]
1/2
Z = 0, if S = 0;
S + 1
⎪⎩ , if S < 0.
[Var(S)]
1/2
n(n − 1)(2n + 5) −
]
g∑
t p (t p − 1)(2t p + 5) ,
where g is the number of tied groups and t p is the number of observations in the
pth group
Once Var(S i ) is computed, we pool across the K seasons and our test statistic
Z is computed.
p=1
If the result of the test statistic for a one sided test is greater
than 1.645 we reject our null hypothesis that no trend exists at the 5% level of
significance.
The seasonal Kendall slope estimator procedure is as follows;
First we compute individual N i slope estimates for the ith season:
Q i = x il − x ik
l − k ,
where x il the data for the i’th season of the l’th year and x ik the data for the i’th
season of the k’th year (l > k).
This process is computed for each of the K seasons. Then rank the N ′ 1 + N ′ 2 +
. . . + N ′ K = N ′ individual slope estimates and find their median. This median is
the seasonal Kendall slope estimator.
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