Re: Gumbel curve fitting

Dougs wrote:
> I am modelling the extreme tail of a distribution, the curve is
> largely normal but is pinched so has elongated tails...
>
> If I fit a Gumbel distribution using the mehtod of moments
estimators
> the fit is very poor.
>
> If I use a method by fitting the distribution at the tail say using
> 0.99,0.991... 0.999 quantiles and a linear regression then the
> distribution fits very neatly. I am not wanting to extrpolate the
> tail very far 0.99999, and the data I have has 50 samples above the
> 0.999 quantile.
>
> What are the pitfalls in this method?
>

This seems OK, but it depends on how picky you or someone else wants
to be:

(a) you seem to be fitting a Gumbel tail - you might want to formalise
this to avoid giving the impression of fitting a Gumbel distribution
to the whole of the population. Some recent work on bivariate
dependence of extremes proceeds by modelling marginal distrubtions at
having a GEV upper tail but just treats the lower portion by using the
sample distribution function over this portion.

(b) if you are using ordinary regression in the tails, you may be
losing some precision of estimation by ignoring the different
variability as the percentage point changes. You might be able to
devise some weighting scheme so as to move to a weighted regression.
Other possibilities are (i) using maximum likelihood for the values
that lie above some threshold: (ii) devise a criterion of fit based on
one of the standrard tests of fit, such as Anderson- Darling, but
limited to the tail of the distribution, then fit by minimising this.
This last has the potential to use "errors" which are "horizontal
errors" as opposed to the "vertical errors" commonly employed for
least squares in this context (ie errors defined as observed value at
given rank minus fitted quantile at given percentage point).

David Jones


.

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