chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
rainfalls in excess of the threshold has a generalized Pareto distribution (GPD),
closely related to the GEV distribution of the early example. The GPD has cumulative
probability function
F(x) = 1 - [1 + ξ (x - T)/σ] -1/ξ ξ≠0
= 1 - exp(-(x - T)/σ), ξ=0
for x greater than the threshold value T=100 mm. The form of F(x) for the
case ξ = 0 is the cumulative probability distribution for the exponential distribution,
which has simpler form than GPD, requiring a single parameter instead of two. The
output from fitting this distribution is given in table 15.3.
Table 15.3. GenStat output from fitting a GPD to daily rainfall data from
Ceres, Argentina, 1944-2002, with threshold value 100 mm.
Threshold = 100
Proportion > Threshold = 0.00143
*** Estimates of GPareto parameters ***
estimate "s.e."
Sigma 13.56 6.055
Eta 0.2638 0.3443
Maximum Log-Likelihood = -89.026
Maximum value of G. Pareto Distribution is Infinite (Eta >= 0)
Significance Test that Eta = 0
(ie pp_1_d_a follows an Exponential distribution)
Likelihood Ratio test statistic: 1.949
Chi-Squared Probability of test: 0.1627
The output of table 15.3 shows that a proportion 0.00143 of daily rainfalls
exceeded the threshold, or 23 events, since the 44-year record contained 16104
daily values. The ML estimates of the GPD parameters σ, ξ ('Sigma', 'Eta') are
13.56 ± 6.055 and 0.2638 ± 0.3443; the value of the shape parameter ξ is less than
its standard error, whilst a formal χ 2 test, shown at the end of table 15.3, shows that
the probability of getting a value of ξ equal to or larger than this value is 0.1627,
not sufficiently small for the hypothesis that ξ=0 to be rejected. Thus, the simpler
exponential distribution can be used to represent daily rainfall exceedances greater
than 100 mm.
A next step might be to test whether there is a time trend in the number of
occurrences of rainfalls in excess of 100mm. The number of occurrences in the 44
years 1959-2002 are 0,0,0,0,1,1,0,2,0,1,0,0,0,0,1,0,0,0,0,0,1,0,1,1,0,2,0,0,1,0,0,1,0,
0,0,1,0,2,1,2,1,0,1,3. These 44 values are well fitted by a Poisson distribution with
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