Dowd - Extreme Value Theory Questions

[mathman]

Hi, I am studying EVT and noticed that the GEV distrib. for xi = 0 is given by: exp[ -exp (x-u/sigma) ] as given in Dowd. So when I translated that as: 1 / (e ^ ( e ^ z) ), I noticed that my solution for ln (p) is coming to : - ( e ^ z) - which is incorrect. As Dowd says that it should be - e ^ ( -z).

In other words:
I also noticed that wikipedia says that the CDF for Gumbel is e ^ ( - e ^ ( - z) ).
But according to Dowd it would be e ^ ( - (e ^ z) ).
So which one is it?

Thanks.

 

[ShaktiRathore]

source wikipedia
If p=exp[ -exp -(x-u/sigma) ]
or p=e^(-e^-z)
or ln p=ln[e^(-e^-z)]
or ln p=-e^-z lne=-e^-z check if something is wrong with exp[ -exp (x-u/sigma) ] as given in Dowd which contrasts with formula given above. check if there is some sign missing or that there is some special case given in Dowd.

thanks

 

[mathman]

Exactly my question. According to Dowd (and schweser notes) it would be e ^ ( - (e ^ z) ). And then they somehow get to the conclusion (after 2 pages) that ln p = - e ^ ( -z). Which contradicts the initial formula (If I haven't made any mistake). And no mention of special case is there.

So I repeat, which one is the correct formula?

Maybe its the fact that x = u + (z * sigma) and x = u - (z * sigma) - I dont know. Thats a far fetched theory from me.

 

[ShaktiRathore]

As per my understanding the as EVT is concerned with the downside of returns that means we are only interested in the formula x = u - (z * sigma) which gives the downside of returns for a mean u and std. deviation sigma of returns. so that x-u=-(z*sigma) or z=-(x-u)/sigma so it should be
e ^ ( - (e ^ z) ).

thanks

 

[mathman]

I thought of that too. But I would want to ask someone who is sure. All the problems have been solved using the other formula : e ^ ( - ( e ^ -z) ). So I think we use that in the exam.

Another question I have: Can someone please explain the graphical interpretation (or even the logical explanation) of how

Peaks over threshold = F( x + u ) - F ( u ) / 1 - F ( u )

(Now I tried to get it and I thought: F (x + u ) = CDF of all x greater than u (u- which is the threshold point) then F ( u ) = Total prob of curve uptil point u. So Numerator = Sum of probability above u. And in the Denominator we have 1 - F (u) which is also the same as total prob. above u. I think I am making some mistake in my interpretation of the variable x here. Can you please help my with this - thanks.)

 

[mathman]

question 3: "Gev requires the estimation of one more parameter than POT" - which parameter are they talking about?

(see if they are talking about the general pot formula - that requires only u (while gev requires - xi, u, sigma), while the var estimate using pot requires 6 parameters - u, beta, xi, n, n(u), and alpha - so what are they talking about?)

thanks.

 

[ShaktiRathore]

Var-POT(u+(β/ξ)*(pow(n*(1-CL)/Nu,-ξ)-1)) which requires many parameters includingβ,ξ(shape parameter) where as the POT distribution has

two parameters.
GEV is

which requires 3 parameters ξ(shape parameter),mu(location),sigma(scale)
you might have got confused by the VaR formula but i think they might be comparing the distributions and the parameters required in them
thanks

 

[aadityafrm]

As z= (x-mu)/sigma, In the generalized expression for the Extreme value distribution as we take the limit as ξ - > 0, the expression inside the bracket will reduce to exp(-z) and hence the whole expression for GEV will become exp {-exp(-z) } (limit of an exponential) .

So, Gumbel distribution cann't be defined as exp(exp(-z)) by ignoring the minus sign. Also to help ans your last question regarding the interpretation of POT expression, hope by rearranging the above expression in F(x+u) -F(u)/1-F(u) as below might help you visualize it -

Pr(X-u<=x|X>u) * Pr(X>u) = Pr(X<=x+u) - Pr(X<=u)
left hand side of the equation is the conditional prob. of the excess losses over a high threshold u multiplied by the prob. of x being over threshold u which makes it logically equal to the area between F(x+u) and F(u).

Let me know if this helps.

 

[mathman]

As z= (x-mu)/sigma, In the generalized expression for the Extreme value distribution as we take the limit as ξ - > 0, the expression inside the bracket will reduce to exp(-z) and hence the whole expression for GEV will become exp {-exp(-z) } (limit of an exponential) .

So, Gumbel distribution cann't be defined as exp(exp(-z)) by ignoring the minus sign. Also to help ans your last question regarding the interpretation of POT expression, hope by rearranging the above expression in F(x+u) -F(u)/1-F(u) as below might help you visualize it -

Pr(X-u<=x|X>u) * Pr(X>u) = Pr(X<=x+u) - Pr(X<=u)
left hand side of the equation is the conditional prob. of the excess losses over a high threshold u multiplied by the prob. of x being over threshold u which makes it logically equal to the area between F(x+u) and F(u).

Let me know if this helps.

Hi Aditya. Thanks for the reply.

So according to you the formula for GEV frechet limits down to e^ (-z) as xi -> 0. I forgot limits, but supposing that is correct, then how does that make the Gumbel equation = e^ ( - e ^ ( -z)) ? (especially since there is only 1 exponential in the gev-fretchet)

And for the the pot formula: I think the reason I am getting confused is because I need to confirm the difference between the following expressions:
So what is the difference between : Pr(X-u<=x|X>u) and Pr(X>u) and {Pr(X<=x+u) - Pr(X<=u)}. graphically?

Thanks again.

 

[mathman]

Var-POT(u+(β/ξ)*(pow(n*(1-CL)/Nu,-ξ)-1)) which requires many parameters includingβ,ξ(shape parameter) where as the POT distribution has

two parameters.
GEV is

which requires 3 parameters ξ(shape parameter),mu(location),sigma(scale)
you might have got confused by the VaR formula but i think they might be comparing the distributions and the parameters required in them
thanks

Hi, thanks for the reply. I did not know that pot is [x/sigma] ^ -alpha. Where is this given?

 

[ShaktiRathore]

In the POT model the excess losses over high thresholds are modelled with the generalized Pareto distribution ?GPD?. This distribution
arises naturally in a key limit theorem in EVT. p,lease refer to the document for more see page 3 ftp://hubble2-1.math.ethz.ch/users/mcneil/cairns.pdf

The most basic GPD is given by the formula [x/sigma] ^ -alpha. src:http://en.wikipedia.org/wiki/Pareto_distribution

thanks