chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
F(x) = exp ( - [1 + ξ(x - µ)/σ] -1/ξ )) ξ≠0
= exp(-exp(-(x - µ)/σ), ξ=0
where the second form with ξ=0 is the well-known Gumbel distribution. We
wish to test whether the location parameter µ changes over the 23-year period of
record, and the simplest starting point is to assume that µ changes linearly over the
period of record: that is, µ = α + βt, where t is the time in years. Thus four parameters
(α, β, σ, ξ) must be estimated from the 23 data values in table 15.1. The
method of fitting is the method of Maximum Likelihood (denoted by ML: see
explanation of the method given below), and it is desired to know whether the
parameter β differs significantly from zero, which would indicate a time-trend. Use
of a standard statistical package (GenStat©) gives the results shown in table 15.2.
Table 15.2. Results from fitting a GEV distribution, with time-trend in the
location parameter µ, to data of table 15.1.
Fitting Trend term: Year
*** Estimates of GEV parameters ***
estimate "s.e."
Mu (Intercept) 30.09 3.190
Sigma 9.017 2.668
Eta 0.2765 0.3223
Slope(Year) 0.05857 0.3079
Maximum Log-Likelihood = -90.498
Maximum value of GEV Distribution is Infinite (Eta >= 0)
Significance Test that Eta = 0
(ie P_01_hr follows a Gumbel distribution)
Likelihood Ratio test statistic: 2.029
Chi-Squared Probability of test: 0.1544
It is seen that the estimate of the slope parameter β is 0.05857 ± 0.3079 (units:
mm year -1 , recalling that the data are one-hour annual maxima), so that the estimate
is much smaller than its standard error. Thus there is no significant evidence that
the location parameter µ of the GEV distribution changed over the period 1975-97.
It is also seen that the shape parameter ξ of the GEV distribution is estimated as
0.2765 ± 0.3223, so that this estimate too is less than its standard error and not significantly
different from zero. This means that the simpler Gumbel form F(x) =
exp(-exp(-(x - µ)/σ), with two parameters instead of three, can be used in further
analyses, instead of the GEV distribution.
The example of the preceding paragraph explored whether the 23 annual
observations showed a trend in time, but it is also possible (again using standard
software) to test whether time-trends exist in the dispersion parameter σ and the
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