Dell’Aquila R., Embrechts P.artax.karlin.mff.cuni.cz/~dvorm3bm/0910z/Vanicek... · 2010. 1....

Extremes and Robustness: Contradiction?

Dell’Aquila R., Embrechts P.

[email protected]

MFF UK

11.1.2010

Vanıcek K. (MFF UK) Extremes and Robustness 11.1.10 1 / 34

Outline

1 Introduction

2 Applying robust methods to EVT

3 Appendix

4 Bibliography


Introduction

Outline

1 Introduction


3 Appendix

4 Bibliography


Introduction

Robust methods in finance

Combining extreme value theory (EVT) and robust methods is not a

contradiction!

Robust methods can provide info on infulential observations, deviatingsubstructures and model mis-specification.

Comprehensive application of different models in risk management,asset allocation and insurance can be found in:

Dell’Aquilla R., Ronchetti E.: Robust statistics and econometrics witheconomic and financial application, New York, Wiley 2006.

EVT practical results about S&P, crisis impact on BAC and otherscan be found in bibliography.


Introduction

EVT-GEV

EVT provides asymptotic distributions of the normalizedmaxima/minima and of the extremes over a high threshold.

For iid random samples we can fit Mn = max(X1, . . . ,Xn) and obtainlimiting distributions of either Gumbel (ξ = 0), Weibull(ξ < 0) orFrechet(ξ > 0).

The limit distribution type and so called maximum domain ofattraction (MDA) can be identified using slowly varying andVon-Mises functions.

All of them can be combined into Generalized Extreme Value (GEV)distribution:

FGEVθ (x) = exp

(

−

(1 +

ξ(x − µ)

β

)−1/ξ)

where θ = (µ, β, ξ)T are location, scale and shape parameters and1 + ξ(x − µ)/β > 0, β > 0.


Introduction

EVT-GEV

If ξ → 0 we obtain FGumµ,β,0(x) = exp(− exp(−(x − µ)/β))

This is appropriate limit of maxima for normal, lognormal, gamma etc.

In the general case for GEV class the density function is given by:

f GEV

(µ,β,ξ)(x) =1

β

(1 + ξ

x − µ

β

)−( 1ξ+1)

exp

(−

(1 + ξ

x − µ

β

)−1/ξ)

and the Gumbel density function is given by:

f Gumµ,β (x) =

1

βexp(−(x − µ)/β) exp(− exp(−(x − µ)/β))


Introduction

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

EVT distributions

x

dgev

(x, x

i = 0

)Gumbel (xi=0)Weibull (xi=−1)Fréchet (xi=1)


Introduction

EVT-GPD

Let X be a random variable with distribution function F and rightendpoint xF . For fixed u < xF

Fu(x) = P(X − u ≤ x |X > u), x ≥ 0,

is the excess distribution function of the random variable X over thethreshold u.The function e(u) = E (X − u|X > u) is called the mean excessfunction.It can be shown that excesses over a high threshold u haveGeneralized Pareto Distribution(GPD):

GGPDξ,β (x) =

{1 − (1 + ξx

β )−1/ξ ξ 6= 0,

1 − exp(−x/β) ξ = 0(1)

where β > 0 and the support is:

x ≥ 0 ξ ≥ 0,0 ≤ x ≤ −β/ξ ξ < 0.


Introduction

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

GPD distributions

x

dgpd

(x, x

i = 0

)xi=0xi=−0.4xi=0.4


Introduction

EVT-GPD

And the density function for GDP (ξ 6= 0):

f GDPξ,β (x) =

1

β

(1 + ξ

x

β

)−( 1ξ+1)

For the case ξ > 0 we obtain heavy-tailed distributions like Pareto,Student, Cauchy, Frechet (tails decay like power functions)

ξ = 0 corresponds to distributions like normal, exponential, gamma,log-normal (tails decay exponentially)

Finally ξ < 0 we obtain distributions with finite right end-point suchas uniform or beta

Parameter θ = (µ, β, ξ)T for both GEV and GPD are typicallyestimated by ML approach ⇒ potential for robust enhancement


Introduction

Robust statistics

M-estimators for some objective function ρ as solution of

θ = arg minθ∈Θ

n∑

i=1

ρ(xi ; θ).

Assume that ρ has derivative Ψ(x ; θ) = ∂ρ(x ;θ)∂θ and then

the M-estimator shall satisfy conditions∑

n

i=1 Ψ(xi ; θ) = 0.

We restrict to Fisher consistent estimators: EFθ[Ψ(X ; θ)] = 0

In terms of ML estimator ρ(x ; θ) = − log fθ(x)

more about M-estimators, Huber functions has already beendiscussed...


Introduction

Robust statistics

Huber function defined as

ψc(r) =

{r if |r | ≤ c ,c · sign(r) otherwise

(2)

Most efficient M-estimator given a bound on the maximal bias isdefined as:

ΨA,ac (x ; θ) = hc (A(θ) (s(x , θ) − a(θ))) ,

where hc(r) = r min(1, c

‖r‖ ) is multivariate version of Huber function

A, a are determined by solving EFθ[hc(A(θ)(X − a(θ)))] = 0 and

EFθ[ΨA,a

c (X ; θ)ΨA,ac (X ; θ)T = I ].


Introduction

Robust statistics

Notice that the M-estimator can be rewriten as:

ΨA,ac (x ; θ) = A(θ)[s(x , θ) − a(θ)]wc(A(θ)[s(x , θ) − a(θ)])

where wc(r ; θ) = min(1, c

‖r‖ ) are weights attached to eachobservation.

For the GPD case the score functions Ψ(x ; θ) = s(x ; θ) are given by

sξ(x ; θ) =x

βξ + ξ2x(ξ + 1) + log

(1 +

ξx

β

)(1

ξ2

)

sβ(x ; θ) = −1

β+

(1 +

ξx

β

)−1

x

(1 + ξ

β2

)


Applying robust methods to EVT

Outline

1 Introduction


3 Appendix

4 Bibliography



Message 1

Robust methods do not downweight ’extreme’ observations if theyconform to the majority of the data.

Consider estimation of a Gumbel model, one of the GEV distributions.

It can be shown that the score function s(x ; (µ, β)) is unbounded in x .

Robust M-estimator for parameters (µ, β) can be found applyingdirectly optimal solution described above.

We would like to highlight that M-estimator can detect data notconforming to the bulk.

We generate 300 observation from 0, 95FGum

(4;2) + 0, 05FGum

(−0,5;0,2).

Both original densities and contamined model are plotted.



0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

Gumbel mu=4,beta=2

x

dgev

(x, x

i = 0

, mu

= 4

, bet

a =

2)

0 5 10 15 200.

00.

51.

01.

52.

0

Gumbel mu=−0.5,beta=0.2

x

dgev

(x, x

i = 0

, mu

= −

0.5,

bet

a =

0.2

)

Figure: Densities of Gumbel distribution with µ = 4, β = 2 on the left andµ = −0, 5, β = 0, 2 on the right.



MLE, PWM and robust Gumbel fit

c

Den

sity

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

PWMTRUEMLERobust

Figure: Illustration of the MLE, PWM and robust estimations of the parametersfor a Gumbel distribution. The classical estimator is attracted by the outlyingobservations around 0.



Next figure shows that classical MLE is strongly attracted by thecontamination.

The robust M-estimator fits the distribution much better where themost data are located.

Additionally it explicitly downweights observations on the left tail.

Most importantly from EVT point of view it does NOT downweightthe observations on the right tail.

Similar robustness issues apply also for GEV, GPD and otherdistributions as well.



Message 2

Robust methods can guarantee a stable efficiency, MSE and a boundedbias over a whole neighbourhood of the assumed distribution.

Consider Pareto model defined by the density

fα(x) = αx−(α+1)xα0 , 0 ≤ x0 ≤ x <∞, α > 0.

Score function is again unbounded in x and MLEα = (1/n

∑n

i=1 log( xi

x0))−1 is not robust.

We generate 300 observations from FPareto(5, 1) and FPareto(5, 10)

For different contaminations 100%ε of the underlying model we plotMSE = E [(θ − θ)2] for MLE and two robust estimators c = (1, 3; 2).

MSE is are averaged over 1000 simulations.



Figure: MSE of the parameter estimates in a Pareto model



It is clearly seen that as contamination increases the MSE for MLEexplodes, while robust estimators are much more stable.

Similar results are obatined for bias and efficiency and for differentdistributions (GPD, Weibull, Gumbel, Gamma, exponential,...).

In practice there is always methodological issue when the ’real’distribution has fatter tails than the assumed one...



Message 3

Robust methods can identify influential points in real data.

Empirical illustration on Danish Fire loss dataset, assumed as iid.

2156 fire insurance losses from 1980 to 1990 inflation adjusted to1985.

Fitting GPD with the threshold u = 10 to 109 remaining excess losses.

In case of classical approach we obtain MLE (ξ, β) = (0, 50; 6, 98)with standard errors (0, 13; 1, 17).

The robust estimates (95% efficiency) are (ξ, β) = (0, 37; 7, 28) withstandard errors (0, 11; 1, 16).

We plot empirical excesses and classical- and robust-fitted GPD.



050

100

150

200

250

Danish Fire loss data > 1 mio Danish Kronen

x@D

ata

1980 1982 1984 1986 1988 1990

Figure: Danish Fire Loss database, threshold 10 mio Danish kronen.



Figure: Empirical distribution of excesses and the classical (solid) and robust(dashed) GPD fits.



Might not be obvious at the first glance, but robust estimatorperforms well for larger values.

It can be also seen that the MLE is attracted by few most extremeobservations and not following majority of data.

This is due to the fact that those points are in fact leverage pointsand robust estimator heavily downweighted them.

We need to make explicit judgementel choice if we want to keep themor get rid of them.

Such decision have to be made on case by case basis in regards to ourinterest and goal.



Message 4

Not all Ψ functions are adequate.

Within M-estimation framework we are free to choose the ψ function,but choice should be made with care.

For example, for the exponential model given byfβ(z) = 1/β exp(−z/β) with β > 0 the MLE is the sample mean.

If we consider M-estimator defined asψRT

b(x) = b(− exp(−x/b) + b/(1 = b)) we obtain robust and

consistent estimator. But we can question if this is appropriate one?

In figure we plot the classical score and ψRTb

(x) for the exponentialmodel with β = 10 and for b = 5, 10.

It can be seen that those functions may deviate significantly alreadyfor central bulk of values.



Figure: Comparison of the classical score function of the exponential model withψRT

b(x) on the left and with the optimal solution on the right.



In comparison optimal solution plotted on next figure follows classicalscore function up to the edge of where most of the data lie, but isbounded for large values of x .

Also the optimal solution has lower maximal bias for the same level ofefficiency and better efficiency for the same maximal bias.

In general different ψ functions may be useful in different situations.



Summary

Message 1

Robust methods do not downweight ’extreme’ observations if theyconform to the majority of the data.

Message 2

Robust methods can guarantee a stable efficiency, MSE and a boundedbias over a whole neighbourhood of the assumed distribution.

Message 3

Robust methods can identify influential point in real data.

Message 4

Not all Ψ functions are adequate.


Appendix

Outline

1 Introduction


3 Appendix

4 Bibliography


Appendix

MDA

Definition (MDA)

We say that random variable X (the distribution function F ) belongs tothe maximum domain of attraction of the extreme value distribution H ifthere exist constants cn > 0, dn ∈ R such that

limn→∞

F n(cnx + dn) = H(x), x ∈ R.

holds. We write X ∈ MDA(H).


Appendix

Slowly varying functions

Definition (Slowly varying functions)

A measurable function g : R+ → R

+ is regularly varying at ∞ with indexa ≥ 0, denoted g ∈ Ra, if

limx→∞

g(sx)

g(x)= sa, s > 0.

We call a the coefficient of variation. If a = 0 we call g(x) slowly varying.Slowly varying functions are usually denoted by L(x).


Appendix

Von-Mises functions

Definition (Von-Mises functions)

Let F be a distribution function with right endpoint xF ≤ ∞. Supposethere exists some z < xF such that F has representation

F (x) = c · exp

−

x∫

z

1

a(t)dt

, z < x < xF , (3)

where c is some positive constant, a(·) is a positive and absolutelycontinuous function with derivation a′(·) and limxրxF

a′(x) = 0.Then F iscalled a von Mises function, the function a(·) the auxiliary function of F .


Bibliography

Outline

1 Introduction


3 Appendix

4 Bibliography


Bibliography

Bibliography EVT

Embrechts P., Kluppelberg C., Mikosch T. (1997) Modelling extremalevents for insurance and finance. Springer-Verlag, Berlin.

Vanıcek K. (2003) Vyuzitı extreme value theory pri rızenı trznıchrizik. Diplomova prace, MFF UK, Praha.

McNeil A., Frey R., Embrechts P. (2005) Quantitative riskmanagement: concepts, techniques and tools. Princeton UniversityPress.

Vesely D. (2009) Teorie extremalnıch rozdelenı ve financıch.Bakalarska prace, MFF UK, Praha.


Dell’Aquila R., Embrechts P.artax.karlin.mff.cuni.cz/~dvorm3bm/0910z/Vanicek... · 2010. 1....

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