Slides alexander-mcneil

Arthur CHARPENTIER, Solvency II’ newspeak

Stress Testing & Reverse Stress Testing

Alexander J. McNeil

[email protected]

http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/

Financial Risks International Forum ‘Risk Dependencies’, March 2010

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Defining halfspace depth

Given y ∈ Rd, and a direction u ∈ Rd, define the closed half space

Hy,u = {x ∈ Rd such that u′x ≤ u′y}

and define depth at point y by

depth(y) = infu,u6=0

{P(Hy,u)}

i.e. the smallest probability of a closed half space containing y.

The empirical version is (see Tukey, 1975)

depth(y) = minu,u6=0

{1n

n∑i=1

1(Xi ∈ Hy,u)

}

For α > 0.5, define the depth set as

Dα = {y ∈ R ∈ Rd such that ≥ 1− α}.

The empirical version is can be related to the bagplot (Rousseeuw & Ruts, 1999).

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Empirical sets extremely sentive to the algorithm

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where the blue set is the empirical estimation for Dα, α = 0.5.

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The bagplot tool

The depth function introduced here is the multivariate extension of standardunivariate depth measures, e.g.

depth(x) = min{F (x), 1− F (x−)}

which satisfies depth(Qα) = min{α, 1− α}. But one can also consider

depth(x) = 2 · F (x) · [1− F (x−)] or depth(x) = 1−∣∣∣∣12 − F (x)

∣∣∣∣ .Possible extensions to functional bagplot.

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The bagplot tool for mortality models

On the a French dataset, we have the following past outliers,

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Age

Log

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19141915

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(here male log-mortality rates in France from 1899 to 2005).

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The bagplot tool for mortality models

Using functional bagplot techniques it is also possible to identify outliers instochastic scenarios,

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Further references

Febrero, N., Galeano, P. & Gonzalez-Manteiga, W. (2007). A functional analysisof NOx levels : location and scale estimation and outlier detection.Computational Statistics 22(3), 411-427.

Hyndman, R.J. & Shang, H.L. (2010). Rainbow plots, bagplots and boxplots forfunctional data. Journal of Computational and Graphical Statistics. 19(1), 29-45.

Rousseeuw, P.J., Ruts, I. & Tukey, J.W. (2009). The bagplot, a bivariate boxplot.American Statistician, 53(4), 382-387.

Sood, A., James, G. & Tellis, G. (2009). Functional Regression : A New Model forPredicting Market Penetration of New Products. Marketing Science, 28(1), 36-51.

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