Re: st: q-q plots, theoretical distribution with values higher than the sample's cutoff point

[David Hoaglin <dchoaglin@gmail.com>]

Nick,

You're correct that, in general, the g-and-h distributions do not have
closed-form densities or cumulative distribution functions.  The
quantile function doesn't exist in closed form either, but only
because the quantile function of the normal distribution is not
closed-form.

For reasons of resistance and robustness, I usually prefer to work
with quantiles.  Fitting by maximum likelihood opens you up to
problems when the distribution has heavy tails and the data may
contain outliers.  Nowadays, fitting a g-and-h distribution by maximum
likelihood is not a major problem, but it is not just a few lines of
code!  I don't know how much has been done on fitting models that
involve predictors.  In any event, the g-and-h distributions are a
valuable part of my toolkit, but not a panacea.

I have no basic problem with maximum likelihood.  I've made heavy use
of it, in Stata and elsewhere.  But good data analysis is iterative:
one should look at data and residuals at various stages.

David Hoaglin

On Fri, Jul 20, 2012 at 10:29 AM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
> Fair question for me at the end. I mean that g- and h- distributions are despite their flexibility rather awkward or elusive customers. It may be just psychology or convenience, but I like distributions with relatively simple closed-form definitions of density, distribution and quantile functions so that I can write a few lines of code to fit them by maximum likelihood, etc. Correct me if I am wrong, but g- and h- don't score well under that heading. As David implies, the practical problem is usually fitting a distribution given predictors, and fitting easily into the ML framework is to me highly desirable.

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