Re: Idempotence and "Replication Insensitivity" are equivalent ?

From: vc <>
Date: 25 Sep 2006 17:09:03 -0700
Message-ID: <> 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

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