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

From   David Hoaglin <>
Subject   Re: st: q-q plots, theoretical distribution with values higher than the sample's cutoff point
Date   Fri, 20 Jul 2012 12:44:51 -0400


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

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 <> 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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