Re: Idempotence and "Replication Insensitivity" are equivalent ?

From: vc <>
Date: 26 Sep 2006 05:52:54 -0700
Message-ID: <> 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

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