# Re: Idempotence and "Replication Insensitivity" are equivalent ?

Date: 26 Sep 2006 02:42:50 -0700

Message-ID: <1159263770.252712.117550_at_h48g2000cwc.googlegroups.com>

vc ha scritto:

*> pamelafluente_at_libero.it wrote:
**> > vc ha scritto:
*

> > > Earlier, I said that the sample median regarded as an interval (50

*> > > percentile) *is* invariant under monotone transformations.
**> >
**> > Ok let's assume that there exists such a thing like a "sample median
**> > regarded as an interval " (it's not a correct expression but I
**> > understand that you want to mean the interval of values which minimizes
**> > the sum of absolute deviations), then you have to define what you mean
**> > by "invariance", as the definition you provided:
**> >
**> > m(f(X)) = f(m(X)).
**> >
**> > does not apply to intervals.
**>
**> I am sorry, I did not realize that you did not understand the above
**> elementary notation. As I said before let 'X' be a real-valued random
**> variable, ' f' a transformation function and 'm' the function mapping
**> the r.v. to a set possibly but not necessarily consisting of a single
**> element and representing the median. Using the function compositionn
**> notation, one could also write m o f o X = f o m o X (which is not
**> much different). Does it help ?
*

You are doing it again.

I do understand the notation. That means

m(f(x)) = f(m(x)) for any x of the support of X

it's "for any x" . Does not apply to intervals. It applies to points.

Further, we are not talking about median of rv's. But about median of a finite set of numerical values. We are not referring to probability measure.

*>
**>
*

> > Only after you have provided such a

*> > definition of invariance, we can check if your statement is ok.
**> >
**> > Otherwise, we are left with a sentence that does not make sense : "
**> > sample(?) median regarded as an interval (50 percentile(?) ) *is*
**> > invariant under monotone transformations ".
**>
**> I hoped you'd realized that, up to now, we'd been talking about what
**> is commonly called 'sample median' or the 'median of a data series' as
**> opposed to the related 'median of probability distribution', but I
**> guess I was wrong.
*

We are not within a sampling framework. There is no need to call for
sampling theory.

>Also, it may be useful to know that sample median

*> *is* the 50th percentile. "Regarded as" can be replaced with "not
**> reduced to a single value when the number of elements is even".
*

That I studied at elementary school. But it's not defined as an interval.

*>
*

> It has as much to do with intervals as the median does because the

*> median is the 50th percentile (or 1/2 -tile). In fact, any quantile
**> can be an interval:
*

"any quantile can be an interval" is not a correct statement.

You are getting mixed up with the theory of countinuous probability
distribution.

We are referring to finite set. For a continuous cdf the median is a
point. For a finite set instead holds the definition I have provided
before.

*>
*

> Let X be a r.v., P denotes probability, then any point 'q' is an

*> R-tile iff it satisfies the two inequalities:
**>
**> P(X <= q) >= R,
**> P(X >= q) >= 1-R
*

Same. We are dealing with finite sets. Leave alone the probability. Received on Tue Sep 26 2006 - 11:42:50 CEST