# [SciPy-User] applications for tukeylambda distribution ?

Friedrich Romstedt friedrichromstedt@gmail....
Tue Nov 1 10:32:55 CDT 2011

```2011/10/30  <josef.pktd@gmail.com>:
> Are there any applications for the Tukey Lambda distribution
> http://en.wikipedia.org/wiki/Tukey_lambda_distribution ?
>
> I just stumbled over it looking at PPCC plots (
> http://www.itl.nist.gov/div898/handbook/eda/section3/ppccplot.htm and
> scipy.stats.morestats) and it looks quite useful covering or
> approximating a large range of distributions.

Hi Josef,

"The most common use of this distribution is to generate a Tukey
lambda PPCC plot of a data set."

"It is typically used to identify an appropriate distribution (see the
comments below) and not used in statistical models directly."
(http://en.wikipedia.org/wiki/Tukey_lambda_distribution)

So I guess the mighty Wikipedia suggests explicitly that there are no
applications in the sense of a statistical model?

Your second reference agrees on this: "The Tukey-Lambda PPCC plot is
used to suggest an appropriate distribution. You should follow-up with
PPCC and probability plots of the appropriate alternatives."

I would guess the "most common use" and "typically used" formulations
are just backdoors to not claim what we're not sure about.

Furthermore, "The probability density function (pdf) and cumulative
distribution function (cdf) are both computed numerically, as the
Tukey lambda distribution does not have a simple, closed form for any
values of the parameters except λ = 0 (see Logistic function).
However, the pdf can be expressed in parametric form, for all values
of λ, in terms of the quantile function and the reciprocal of the
quantile density function."
(http://en.wikipedia.org/wiki/Tukey_lambda_distribution again); I have
no idea off the cuff if it is useful to fit the quantile directly or
not.  At least it does not look like the common (aside of PPCC).

I guess the reason why one would not like to model with this Tukey
quantile is, that it is just a parametric model, and does not have a
physical reason (at least there is none such given in what you gave).

A quick googling "tukey modeling" gives this here:
http://andrewgelman.com/2011/01/tukeys_philosop/ - but I find what's
written there rather unclear and don't really understand what're the
associations related to "model" and "method" in that post.  It looks
like if it comes down to "do we need a physical derivation of our
distribution or not".

To my belief, a pdf or cdf without a reason is missing something.  It
just feels wrong - it's not satisfying.  The problem might be: Where
does the endless circle of deriving and deriving stop?  Maybe it never
does.  Maybe it's just a game we play and we pretend that it's of
objective importance but it isn't - it's just about satisfaction and
fun in the end. :-)

AISI,
Friedrich
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