Analysis of Trends in Low-Flow Time Series of Canadian Rivers - Hydrologic Time Series Analysis: Theory and Practice - page 203

10.2.3 Assumptions about Serial Structure: Independence vs
Short- and Long-term Persistence
It is documented in the literature that the performance of the MK test is
seriously affected by serial structure of the time series being tested, i.e. if a
time series is positively (negatively) correlated then the MK test will suggest
a significant trend more (less) often than it will for an independent series (von
Storch, 1995). The serial structure of a hydrological time series could exhibit
short-term persistence (STP) or long-term persistence (LTP) or no persistence
at all. In the STP case, it may resemble to serial structures of autoregressive
moving average (ARMA) type stochastic process (Box and Jenkins, 1970)
and in the LTP case, it may resemble to serial structures of fractional Gaussian
noise or fractional autoregressive integrated moving average (FARIMA) type
stochastic process (Hosking, 1984). To address the influence of STP on trend
significance, Hamed and Rao (1998) and Yue and Wang (2004) proposed two
different modified MK tests (respectively referred to as MMK1 and MMK2
hereafter) based on the effective sample size approach, originally introduced
by Bayley and Hammersley (1946) and implemented by Lettenmaier (1976)
to develop the modified SROC test of trend. Since the influence of LTP on
trend significance is considerably different from that of STP, Cohn and Lins
(2005) proposed adjusted likelihood ratio test (ALRT) based on FARIMA-
type time series modelling and simulation approach that was further elaborated
in Khaliq et al. (2009b). In addition, Hamed (2008) proposed another modified
test, MK scaling (MKS) test, to address the influence of LTP on trend
significance.
The influence of STP on the performance of the MK test is demonstrated
in Fig. 10.2, using hundred thousand samples generated from an AR(1) process
Fig. 10.2. Rejection rates of (a) MK, (b) MMK1, and (c) MMK2 tests for eight
sample sizes ranging from 30 to 100, with an interval of 10. The inset in Fig.
10.2(a) is an enlarged, off scale, view of the rejection rates of the MK test. The
dotted line corresponds to the nominal significance level (0.05). Autocorrelations
of data were used in Fig. 10.2(c) and those of their ranks in Fig. 10.2(b). Figure
adopted and modified from Khaliq et al. (2009b).