chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
at sites 100 km apart. To allow for this spatial correlation, more sophisticated models,
analogous to the kriging models used where spatial data are Normally distributed,
have been proposed by Casson and Coles (1999). These authors develop a
spatial model that is also based on GEV distributions at each site, but with a spatial
'latent process' to describe the variability in µ and other parameters, which are now
regarded as random variables. The stochastic process used to model µ(z), where z
is now used to describe the two or three spatial co-ordinates, is
hµ = (µ(z)) = fµ (z; βµ) + Sµ (z; αµ)
with similar expressions for σ(z) y β(z). Fitting them requires the use of Monte
Carlo Markov Chain (MCMC) iterative procedures, which are revolutionizing the
potential for undertaking calculations in multi-dimensional space.
15.7. Methods based on L-moments
The use of sample moments (mean, variance, coefficients of skewness and
kurtosis) to provide statistical summaries of data sets is well known. In addition to
their use as numerical summaries, these sample moments were also used in the past
(before desk-top computers became widely available) to estimate the parameters of
probability distributions, such as the GEV mentioned above; the mean, variance
and skewness of the probability distribution to be fitted were simply equated to
their values calculated from data, and the resulting equations were solved, by iterative
methods when necessary, to give estimates of the distribution parameters. This
estimation by the “Method of Moments” (MM) often avoided the numerical complexity
of Maximum Likelihood (ML) estimation, but is theoretical inferior to ML
estimation, for reasons presented in statistical texts. One of the reasons contributing
to the inferiority of MM estimation is that the variance and skewness coefficients
calculated from data are subject to large sampling errors.
As a better alternative to the use of sample moments, Hosking and his fellowworkers
have proposed the use of probability-weighted moments, from which Lmoments
are calculated. These L-moments have more advantageous properties than
variances and coefficients of asymmetry and kurtosis, and using them to fit probability
distributions is often simpler than fitting by ML. There is now an extensive
literature (Hosking et al. 1985; Hosking and Wallis 1997) on the use of L-moments
in the analysis of hydrological extremes. The book by Hosking and Wallis (1997)
in particular gives a very thorough account of the use of L-moments for regionalization
of hydrological variables. The authors give methods for estimating GEV
parameters (as well as the parameters of other distributions) by L-moments as an
alternative to the more complex calculations required for the calculation of ML estimates.
It is certainly true that L-moments can be calculated in some cases where
ML estimation procedures fail, for example when the ML iterative calculations fail
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