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

From: <pamelafluente_at_libero.it>
Date: 26 Sep 2006 02:42:50 -0700

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.

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

```

>
>
> > 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.

>
> 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

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