Extreme Events – Specimen Question
A.4.1(b) – Answer/Hints
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Q. Do these evolving
estimates of c appear to be stable? How would you test such an assertion
statistically?
The estimates derived in A.4.1(a)
do not appear to be stable – early ones are generally negative, whilst later
ones are generally positive.
Whilst it is possible to create analytical statistical tests
for many problems, it is often easier to carry out a Monte Carlo simulation, in
which we simulate the outcomes assuming that some prior model is correct and we
work out the proportion of times that outcomes as extreme as observed outcome
arise in the simulation. This, of course, still requires us to identify
significance levels etc. as would be the case with any other type of
statistical technique
Leaving aside generic issues to do with simulation
techniques (such as numbers of simulations to carry out, see e.g. Section 6.11
of the book Extreme
Events), the main challenges with applying such a methodology to this type
of problem are:
(a) Defining
the right prior distribution and adjusting the problem to take account of
degrees of freedom introduced by parameter estimation. In this particular case
the form of the prior is well defined, but there is flexibility over the
selected value of c. We cannot assume, say, that the ‘true’ model
involves c = 0.3218388. This value was itself estimated. So instead, we
might carry out simulations as if c = 0.3218388 but then include an
adjustment to the elements of each separate simulation forcing the results
always to correspond to this value (in effect a ‘constrained’ simulation).
Imposing a constraint in this manner can be done in several different ways,
each of which is implicitly adjusting somewhat the prior distribution we are
implicitly using in our testing, so we need to take this into account in our
end conclusions
(b) Defining
how to measure how far away from the ‘expected’ are the actual observations.
This problem is a generic one whenever we have several different observations
within the overall observation set. We need to take a view on whether we are
most interested in the spread of differences, the most extreme difference etc.
Some of the issues are explored further in pages on the website relating to tests for normality.
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