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### Extreme Events – Specimen Question A.2.3(b) – Answer/Hints

Q. Prepare a (standardised) quantile-quantile plot of the weighted data. Is it also suggestive of fat-tailed behaviour? Hint: the ‘expected’ values for such plots need to bear in mind the weight given to the observation in question.

We need to:

(a)    Scale the weights so that they add to unity

(b)   Standardise the observations, so that they have weighted mean equal to zero and weighted (sample) standard deviation equal to unity

(c)    Sort the data

(d)   Work out the quantile plots corresponding to each data point (e.g. assume the data point corresponds to the cumulative (scaled) weight of smaller points plus one-half of the (scaled) weight of the observation in question

(e)   Identify standardised inverse normal values corresponding to the quantile points in (d)

(f)     Plot ‘observed’ vs ‘expected’, the latter being the values from (e)

Steps (a) and (b) give:

 Period Scaled Weight Standardised Observation 1 0.0239245 -3.7746674 2 0.0256416 0.0229646 3 0.0274820 2.8477710 4 0.0294545 1.0055183 5 0.0315686 -0.1747424 6 0.0338343 0.0751858 7 0.0362627 -0.0689278 8 0.0388654 0.1401715 9 0.0416550 0.9325003 10 0.0446447 -1.6640468 11 0.0478490 0.1012191 12 0.0512833 0.1142167 13 0.0549640 -0.4974175 14 0.0589090 0.3714857 15 0.0631371 -0.5656546 16 0.0676687 -0.1482089 17 0.0725255 -0.9404055 18 0.0777309 0.8345045 19 0.0833099 0.4729935 20 0.0892893 0.2434824

Steps (c), (d) and (e) give:

 Scaled Weight Observation (i.e. ‘Observed’) Quantile Point ‘Expected’ 0.0239245 -3.7746674 0.0119622 -2.2583397 0.0446447 -1.6640468 0.0462468 -1.6823879 0.0725255 -0.9404055 0.1048319 -1.2544904 0.0631371 -0.5656546 0.1726632 -0.9436933 0.054964 -0.4974175 0.2317138 -0.7332146 0.0315686 -0.1747424 0.2749801 -0.5978199 0.0676687 -0.1482089 0.3245987 -0.4548775 0.0362627 -0.0689278 0.3765644 -0.3145165 0.0256416 0.0229646 0.4075166 -0.233938 0.0338343 0.0751858 0.4372545 -0.1579336 0.047849 0.1012191 0.4780962 -0.0549323 0.0512833 0.1142167 0.5276623 0.0693948 0.0388654 0.1401715 0.5727367 0.1833459 0.0892893 0.2434824 0.6368141 0.3499558 0.058909 0.3714857 0.7109132 0.5560546 0.0833099 0.4729935 0.7820227 0.7790426 0.0777309 0.8345045 0.8625431 1.0918164 0.041655 0.9325003 0.922236 1.4202738 0.0294545 1.0055183 0.9577907 1.7256048 0.027482 2.847771 0.986259 2.2046022

The following chart plots the observed vs expected values, using Microsoft Excel. It is also suggestive of fat-tailed behaviour:

However, in practice it is easier to use the Nematrian Charting facility, i.e. MnPlotWeightedStandardisedQQ which can do all of these steps simultaneously.