Changing Extremes
21st February 2016
The MRes students of STOR-i are currently in the middle of our short research topic projects. These involve writing a
literature review into one of the topics presented to us a few weeks ago. Some of these were discussed in
Optimising with Ants and Ice-cream Cones,
Can Hamiltonian Win At Monte Carlo? and Gaussian Processes.
The topic I have chosen to look at in more detail is an extension of the subject of
Tails, Droughts and Extremes, that is Non-stationary Extremes. This was presented by
to ozone data, where the
location parameter of a GEV was assumed to follow a DLM. They showed that this approach is that any trend, whether
short or long term, can be picked out without assuming a parametric form, making it very flexible. However, there
are many parameters to estimate which may involve a lot of computational time.
Varying Extreme Parameters with Covariates
This is by far the most common method I have come across, and there are multiple ways of going about it. Unlike the
DLM approach, it is more common to use a GP distribution, in which case the choice of $u$ is key. One way is to
split the data into predefined "seasons" that are assumed approximately stationary and then select $u$ such that
the exceedance probability, $p$, is constant. If there are more covariates involved than just time, one could allow $u$ to
vary with the covariates. This was used by suggest a
Box-Cox method of the form
$$\frac{Y_t^{\lambda_t}-1}{\lambda_t} = \mu_t + \sigma_tZ_t.$$
Once $Z_t$ has been obtained, the extremes of $Z_t$ can be analysed in a more straight forward manner. The benefits
of this approach are that the non-stationary behaviour can be estimated much more reliably than in the previous
methods. In addition, it has more statistical efficiency. On the down side, if the extremes of $Y_t$ do not follow
the same non-stationarity as the body, then $Z_t$ will require another non-stationary method. The other negative
is that it is much more complicated than the methods above.
All of these methods are interesting, and have been very interesting to read about. There does seem to be a lot of
disagreement as to which method is best, but I imagine that it is which method is the best depends very much on the
situation and application.
References
[1]
Chavez-Demoulin V., Davison A. C., REVSTAT – Statistical Journal, Volume 10, Number 1, pp.109–133
(2012).
[2]
Gabriel Huerta, Bruno Sanso, Environ Ecol Stat, 14, pp.285–299
(2007).
[3]
P. J. Northrop and P. Jonathan, Environmetrics, Vol.22(7), pp.799-809
(2011).
[4] -->
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