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

Date: 26 Sep 2006 05:52:54 -0700

Message-ID: <1159275174.847727.235780_at_b28g2000cwb.googlegroups.com>

pamelafluente_at_libero.it wrote:

*> vc ha scritto:
*

[...]

*>>
**>
*

> 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.
*

' m(f(x)) = f(m(x))' is a standard and very simple definition of quantile invarince under monotonic transformations that can be found in any statistics course.

*>
*

> For intervals for instance you could define invariance something that

*> leave constant the probability measure of the interval or whatever. Or
**> (another definition) you could say that the above must hold for each x
**> in the interval. But if one takes this second definition, you have
**> already proven that invariance does not hold.
**>
**> So you still need to define invariance for intervals, and I have
**> already warned you about the pointwise definition.
*

The passage above does not make any obvious sense. The only interval we are talking about here is the median itself.

*>
*

> Further, we are not talking about median of rv's. But about median of a

*> finite set of numerical values.
*

The expression "median of a finite set of numerical values" does not make any sense whatsoever unless such set is a random sample realization of some observations/experiment.

[...]

> >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.
*

Apparently, your elementary school education was severely lacking.

*>
**> >
*

> > 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.
*

The fact that any quantile can be an interval is a simple consequence of the quantile definition that can be found in any introductory probability textbook (see my previous message re. the definition).

*>
*

> 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.
*

It may come to you as a surprise, but a random sample realization of size n represents n random variables with the same probability distribution which can be either continuous or discrete. Also, as I said earlier, talking about a median definition "for a finite set" is meaningless unless such set is a random sample realization.

*> >
*

> > 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.
*

"

talking about a median definition "for a finite set" is meaningless
unless such set is a random sample realization.
"

I am afraid that in this message you've shown that your understanding of elementary probability notions is very superficial. Received on Tue Sep 26 2006 - 14:52:54 CEST