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### Extreme Events – Specimen Question A.4.1(b) – Answer/Hints

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.