| The widely-used hydrological procedures for
calculating events with T-year return periods from data that follow a
Gumbel distribution assume that the data sequence from which the Gumbel
distribution is fitted remains stationary in time. If non-stationarity is
suspected, whether as a consequence of changes in land-use practices or climate,
it is common practice to test the significance of trend by either of two
methods: linear regression, which assumes that data in the record have a Normal
distribution with mean value that possibly varies with time; or a non-parametric
test such as that of Mann-Kendall, which makes no assumption about the
distribution of the data. Thus, the hypothesis that the data are Gumbel-distributed
is temporarily abandoned while testing for trend, but is re-adopted if the trend
proves to be not significant, when events with T-year return periods are
then calculated. This is illogical. The paper describes an alternative model in
which the Gumbel distribution has a (possibly) time-variant mean, the time-trend
in mean value being determined, for the present purpose, by a single parameter β
estimated by Maximum Likelihood (ML). The large-sample variance of the ML
estimate ˆβMR is compared with the variance
of the trend βLR
calculated by linear regression; the latter is found to be 64% greater.
Simulated samples from a standard Gumbel distribution were given superimposed
linear trends of different magnitudes, and the power of each of three
trend-testing procedures (Maximum Likelihood, Linear Regression, and the
non-parametric Mann-Kendall test) were compared. The ML test was always more
powerful than either the Linear Regression or Mann-Kendall test, whatever the
(positive) value of the trend β; the
power of the MK test was always least, for all values of β.
Keywords: Extreme value probability distribution, Gumbel distribution,
statistical stationarity, trend-testing procedures |