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

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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.

 


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