Attached (in a spreadsheet) is a potential Ad hoc tail estimator for use with t distribution. The idea behind estimator is
 to calculate a standard Hill tail estimator
 calculate a second estimator that is excess over normal of tail but other wise same simple calc as Hill
 blend these estimators via some ratio so that the excess estimator dominates if Hill estimator is small (this means estimator will come out at near zero for a normal distribution)and Hill estimator dominates if Hill estimator is large
It has to use an ad hoc ratio for which i cannot justify but it looks like it might work reasonably.........
Risk using Student T
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 Joined: Sun Sep 28, 2008 10:30 pm
Re: Risk using Student T
Is that the ETL/VaR estimator? What about skewness  or are you using symmetric t?
Re: Risk using Student T
Am not using the ETL/VAR estimator. It turns out that estimator would be very similar to standard Hill Estimator in that it would use exactly same data points as input but use average of ratios over threshold rather than average of ln of ratios over threshold used in Hill.
So the measures are very related and very approximately ETL/VAR= 1/(1Hill)
I think Hill though is more stable as a basis for an estimator so probably better.
For data analysis intend to just look at one tail at a time. So i think this means this estimator would work for input into either  and as would potetially give two outputs one for each tail maybe suited to non symetric T (although i have never used that)
So the measures are very related and very approximately ETL/VAR= 1/(1Hill)
I think Hill though is more stable as a basis for an estimator so probably better.
For data analysis intend to just look at one tail at a time. So i think this means this estimator would work for input into either  and as would potetially give two outputs one for each tail maybe suited to non symetric T (although i have never used that)

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Risk using Student T
Want me to find some references to nonsymmetric t?
Re: Risk using Student T
No that is ok you are too busy for that and that is not my focus now.
More interested on any views on the tail estimator in the attached speadsheet it gets around I think the "Hill Horror Plot" shown in Dowds book P199 and makes output a suitable input suitable for T distibution
More interested on any views on the tail estimator in the attached speadsheet it gets around I think the "Hill Horror Plot" shown in Dowds book P199 and makes output a suitable input suitable for T distibution

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Risk using Student T
Hiya
Well, your spreadsheet seems to work well on the data you are giving it. You have a nice 'full' data set with 2000 obs in your experiment. What if you bootstrapped a smaller sample from this, say with only 500 data points  what level of accuracy do you have now for the d.f. estimate?
Well, your spreadsheet seems to work well on the data you are giving it. You have a nice 'full' data set with 2000 obs in your experiment. What if you bootstrapped a smaller sample from this, say with only 500 data points  what level of accuracy do you have now for the d.f. estimate?
Re: Risk using Student T
Hi tks for your comments/help. I am going to test it as a real estimator rather than as a theoretical estimator soon (a) against randomised T distribution data (b) real data. All these estimators have quite a degree of variance when used in practice so i think you have to apply to lots of different sets of data and take a sort of average outcome.I am quite hopeful that it will provide a good practical solution that will have less cut off problems and less bias when using as input to T. Be good if there was more theoretical justification for it.
Did you say that you had a formula for ETL/VAR ratio for t distribution in the new book. And if so any chance of a preview of the formula on the forum.
Tks
Did you say that you had a formula for ETL/VAR ratio for t distribution in the new book. And if so any chance of a preview of the formula on the forum.
Tks
Re: Risk using Student T
Carol, Thank you very much for that. Of course I will be buying the book which i am looking forward to. Cheers
Re: Risk using Student T
I have been busy and not much time to focus on this recently but one observation that i find interesting is the quite similar gradient of slope of ETL/VAR as you move from say 90% to 99% cut off for T distribution with different DF  always downwards sloping (ie 90% > 99%) and almost linear and gradient looks very silmilar despite DF (ie gradient similar to normal). Obviously it is not linear across full space because in very far tail it flattens out to be = DF /(DF1)
Also by observation alpha stable distribution empirically has slope other way for low alpha ( ie 90% less than 99%) i think but must switch as at some point it also converges on normal as alpha goes to 2  difficult to analyse as we lack analytical CDF i think for alpha stable
GPD can do both upward and doward sloping and maybe this is one way to think of difficult to understand Beta parameter it is this parameter taht controls this slope before you get to very far tail (whereas tail shape/tail index param controls the limiting ratio in very far tail)
A pure Power law (special case of pareto) has a flat gradient for ETL/VAR for diiferent percentiles.
[As we have previously noted on this thread there is vclose relationship between Hill estimator and shape of ETL/VAR]
Also by observation alpha stable distribution empirically has slope other way for low alpha ( ie 90% less than 99%) i think but must switch as at some point it also converges on normal as alpha goes to 2  difficult to analyse as we lack analytical CDF i think for alpha stable
GPD can do both upward and doward sloping and maybe this is one way to think of difficult to understand Beta parameter it is this parameter taht controls this slope before you get to very far tail (whereas tail shape/tail index param controls the limiting ratio in very far tail)
A pure Power law (special case of pareto) has a flat gradient for ETL/VAR for diiferent percentiles.
[As we have previously noted on this thread there is vclose relationship between Hill estimator and shape of ETL/VAR]
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