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Re: Idempotence and "Replication Insensitivity" are equivalent ?

From: vc <boston103_at_hotmail.com>
Date: 25 Sep 2006 08:19:04 -0700
Message-ID: <1159197544.312807.105790@i3g2000cwc.googlegroups.com>

pamelafluente_at_libero.it wrote:
> vc ha scritto:
> > [...]
> > > In order to have a unique solution (and therefore equivalence), in the
> > > case we are dealing with finite set of numbers when the cardinality is
> > > not odd, the definition of MEDIAN is "completed" by *assuming* (it's
> > > an ASSUMPTION, a convention) as median the average of the 2 central
> > > terms of the ordered sequence of numbers:
> > >
> > > MEDIAN = ( x(n\2) + x(n\2 + 1) ) / 2
> >
> > The convention, while convenient in many cases, does not work if we
> > want to preserve invariance to monotonic transformations. Considering
> > the entire interval as the median is the only way to preserve the
> > invariance property.
>
> I have simply recalled the standard definition,
> but I have some "feeling" that this can be an interesting point.
>
> Can you make an example of what you have exactly in mind?

Consider for example logarithmic(base 10) measurement scaling: (10, 100, 1000, 10000) -> (1,2,3,4). Or income adjusted for inflation, or any kind of data transformation.

> Stated so generally it's not clear (to me) what you are referring to.
> Invariance is sometimes used with different meanings,
> eg, in the case of the variance we say that it is
> translation-invariant.
> What kind of "invariance" have you in mind ?

Let X is a randon variable, f is a transformation function and m is the median. Then median invariance would mean that m(f(X)) = f(m(X)). Received on Mon Sep 25 2006 - 10:19:04 CDT

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