Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 68

( i = 1, 2, …, n -1) and summed up. The total number of inversions denoted by
I has a mean (P I* ) and variance (

I V ) given as follows (Shahin et al., 1993):

2
*

nn 1

P I* = (57)

4

3 2

235
72

nn n

I V =

and (58)

As n the approximate standard normal variate ( z ) can be computed ,

as:

*

I P
V

I*

z = (59)

I*

If the computed z -value is within the acceptable range, i.e., ±1.96, which
are the critical values for the two-sided test at 5% significance level, the null
hypothesis that the time series is trend-free cannot be rejected.

4.3.12 Kendall's Rank Correlation Test

The Kendall's rank correlation test is mostly preferred for trend detection in
hydrologic time series (Jayawardena and Lai, 1989; Zipper et al., 1998; Kumar,
2003). In this test, a null hypothesis of no trend is initially assumed, and then
the test is carried out to reject or accept the hypothesis. If a series x t ( t = 1, 2,
..., n ) is to be examined, the number of times ( p ) that x j > x i is counted in all
pairs of observations ( x i , x j ). The Kendall's test-statistic (W) is defined as
follows:

p
nn

4 1
( )

W = (60)

The test-statistic is then expressed as a standard normal variate in the
following form (Kendall, 1973):

W

z = (61)

Var( ) W

2(2 n

nn

5)

where Var(W) = (62)

9( 1)

If the value of ' z ' lies within the limits ±1.96 at the 5% significance level,
the null hypothesis of no trend cannot be rejected.

4.3.13 Mann-Kendall Test

This is a nonparametric test for exploring a trend in a time series without
specifying the type of trend (i.e., linear or nonlinear). Mann (1945) originally