Extreme Events – Specimen Question
A.2.1(d) – Answer/Hints
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Q. Does the Cornish-Fisher
4th moment approximation appear to under or overstate the fat-tailed behaviour
of this series?
The Cornish-Fisher 4th moment approximation, see MnCornishFisher4, in
effect characterises the form of the QQ-plot by a cubic, which if the sample
data has been standardised (using e.g. MnNormaliseArray)
involves a plot of the following form:
Whilst the series underlying this plot can be created
relatively simply in, say, Microsoft Excel (or using MnCornishFisher4, it
is again probably simpler to use the Nematrian website charting capability that
makes the creation of such a chart very simple, using MnPlotStandardisedQQWithComparisons.
The Cornish-Fisher 4th moment approximation appears in this
instance to be providing a reasonable representation of the distributional form
towards the tails of the distribution, if anything perhaps understating the
extent of fat-tailed behaviour.
This is in contrast to its apparent tendency to overstate
fat-tailed behaviour in some of the index series analysed in the book Extreme Events. The last
observation in either tail in the small sample analysed here is actually only at
the 2.5%/97.5% quantile level, so is not particularly far into the tail
relative to some of the larger samples analysed in that book.
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