Trying to predict extreme values corresponding to rare events I need some advice on methodology.

I need to predict a numeric value (claim amount) being as good as possible on high values corresponding to rare events (corporal damage, technological disaster...).

Any suggestion on methodology would be appreciated. Thanks.
My suggestion:

There is a technique called Quantile Regression, which allows you to determine the highest (or lowest) values of a variable. This is valid only if you have regressors (exogenous variables).