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Re: Idempotence and "Replication Insensitivity" are equivalent ?

From: <pamelafluente_at_libero.it>
Date: 26 Sep 2006 08:34:09 -0700
Message-ID: <1159284848.989206.107640@d34g2000cwd.googlegroups.com>


vc ha scritto:

> pamelafluente_at_libero.it wrote:
> > vc ha scritto:
> >
> > > pamelafluente_at_libero.it wrote:
> > > > vc ha scritto:
> > > ' m(f(x)) = f(m(x))' is a standard and very simple definition of
> > > quantile invarince under monotonic transformations that can be found in
> > > any statistics course.
> >
> > I know what is meant by saying that taking the interval between the 2
> > central values is a way to preserve invariance wrt to monotonic trans,
> > and I do agree with that, but.. the point is that you do not seem to be
> > aware of the meaning of that statement
> >
> > Tell me what it means to you that an *Interval*, such as the median
> > values, is invariant wrt to monotonic transf. Let's make an example
> > with:
> > 10 100 1000 10000 and Log. What does it mean to you that the median
> > interval [100, 1000] is invariant wrt to Log transformation and how do
> > you fit it in the expression m(f(x)) = f(m(x)) ?
>
> log(X): {1, 2, 3, 4}, m(log(X)): [2, 3]
> m(X) : [100, 1000], log(m(X)) : [2, 3]

Ah finally. That's just wanted you to get aware of: You say:

m(X) : [100, 1000],

therefore log(m(X)) is formally the same as log( [100, 1000] )

what does log( [100, 1000] ) means ?

Nothing. If you do not define it.

So the answer was that what actually is invariant is not the interval (which contains point that are not invariant, as you have shown), but the end points. As you see you needed to define invariance, as your definition:

 ' m(f(x)) = f(m(x))'

does not apply here but needs adjustments.

>
> > Descriptive statistics, and the median concept exist independently of
> > the notion of probability measure (where they get generalized).
>
> Sorry but that is obviously a nonsensical statement. Statistics is not
> a collection of tricks that one can apply to data and come up with an
> answer. Learning probability is a precondition to understanding
> statistical methods which otherwise may look as a meaningless number
> game. First, one starts with the notions like a random experiment,
> sample space S and probability P; then, one moves to the idea of
> random variable, a random sample of size n, and other statistical
> stuff. Your elementary school curriculum may be different though.
>

 I do not agree with that. But I am not going to argue with this opinion.

 I just note that MEDIAN() is used by every DBA or user and they need to
 know nothing about probability measure.

> >Of course any set of distinct values can be seen as a uniform discrete
> > distribution, but that is not necessary.
>
> That statement does not make any obvious sense.

 It does to me. {1 2 5} can be seen as a uniform with masses equal to 1/3.

 Here again our opinions are not coincident.  

 Thanks for the instructive discussion :)

-P Received on Tue Sep 26 2006 - 10:34:09 CDT

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