Tail Fitting of a Normal Distribution
[this page | pdf | back links]
One approach for fitting the tail of a distribution is to
select an appropriate distributional family and then to select the parameters
characterising the distribution in a manner that provides the best fit of the
observed (cumulative) distribution function and/or quantile-quantile plot in
the relevant tail.
Suppose that the observations are for
When ordered these are say .
Weights given to each observation in the curve fitting process are .
Typically we might expect the to
be non-zero (and then typically constant) only for sufficiently
small, or for sufficiently large, although
this is not strictly necessary.
A common way of carrying out curve fitting is least squares,
so a natural way of implementing this approach to fit a (univariate) Normal
distribution to the data might be:
Any Normal distribution is characterised by a mean, , and standard
deviation, . We might therefore derive, ,
the expected value for the observation ,
using the following formula:
[Note, the expected value of ’th quantile of
a Normal distribution is not precisely as
defined above because the pdf is not flat, see e.g. Expected Worst
We would then identify estimates of the mean, ,
and standard deviation, ,
that together minimise the following least squares computation:
This is minimised when and
i.e. for the values of and
these equations simplify to:
Whilst this type of approach is primarily designed to be
used merely in the tail of the distribution (i.e. with non-zero,
perhaps constant, only for suitably small or, for the other
tail, only for suitably close to ), we can also
consider what answer this approach would give if it were applied to the entire
distributional form, e.g. using for
all . As the are
symmetric around 0.5, we have so
then the usual maximum likelihood estimator .
By, say, carrying out a simulation exercise we can also confirm that is
also typically close to the relevant maximum likelihood estimator if is not very