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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