# [SciPy-User] applications for tukeylambda distribution ?

josef.pktd@gmai... josef.pktd@gmai...
Tue Nov 1 11:11:28 CDT 2011

```On Tue, Nov 1, 2011 at 11:32 AM, Friedrich Romstedt
<friedrichromstedt@gmail.com> wrote:
> 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).

That's pretty much the impression that I also got. However, we do have
the cdf and indirectly the pdf in scipy.special. It also has a section
in Johnson, Kotz and Balakrishnan. So, I was wondering whether it's
used in any field.

>
> 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. :-)

The Gellman page is a bit too philosophical for my taste, especially

To some extend I'm just a collector (of statistical functions instead
of coins or movies or powertools), but I'd rather collect useful
things (you never know when they come in handy).

My impression is that in reliability they invent about ten
(underestimate) new distributions a year that all have a motivating
introduction (that doesn't tell me much but looks relevant) (bath-tub
shapes, anyone?:)

Thanks for looking into it.

Josef
too many brackets

>
> AISI,
> Friedrich
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