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

Date: 25 Sep 2006 17:09:03 -0700

Message-ID: <1159229343.195135.100820_at_b28g2000cwb.googlegroups.com>

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 ?

> 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. 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".

*>
**>
**> -P
**>
**> PS
*

> The 50 percentile has nothing to do with 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:

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
Received on Tue Sep 26 2006 - 02:09:03 CEST