Re: Looking for an Introduction to Statistics


"Herman Rubin" <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote in message
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In article <i_qdnXNVxtjzVKzbnZ2dnUVZ8tSdnZ2d@xxxxxx>,
Nick <tulse04-news1@xxxxxxxxxxx> wrote:

"Herman Rubin" <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote in message
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In article <1177541307.943641.324560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
tactics <tactics40@xxxxxxxxx> wrote:
I'm wondering if there are any well-written books that explore
Statistics from the viewpoint of Set Theory? I find statistics
interesting, but I can't stand the heavily emphasis on Applied
Statistics and I would rather have an introduction that builds an
understanding of the subject in the language of sets.

For all x, one should not learn to apply x unless
one is familiar with the theory of x.

Does anyone know of any such book?

I do not believe that there is one which explicitly
uses sets, except through probability. Now probability
certainly uses sets. I do not believe anyone should
encounter statistics without at least an approach to
measure-theoretic probability.

I have studied probability and statistics, and I seem to remember also
doing
something on measure theory in my maths degree. But I never recall using
the
term "measure theory" wrt probability (although I do see that there are
lot
of such references).

What is probability except measure with the whole space
having measure 1? Of course, this restriction allows one
to do quite a bit more, and there are "real-world intuitions"
connected with probability which do not work for general
measures. One can define conditional measure, and it always
gives probability.

Now it is not necessary to know the proofs of the measure
theory theorems to use them, but the ideas of measure theory,
in particular those forced by countable additivity, are of
great importance. Finitely additive measures, and finitely
additive probability, lead to MAJOR problems.

It would seem to me that for someone who is clearly approaching the
subject,
using such a term seems to be a bit blinding with science (IMHO).

If I had to look up the term I wondered what the OP would make of it, who
seems to just approaching the subject.

Unfortunately, the usual treatments hide the intuitions.
Instead of starting with length, start with counting,
and then add more general discrete measures, such as
weight. The earliest use of integration was when some
merchant computed his bill as so many bottles of wine
at so many shekels per bottle, so many weights of wheat
(a weight need not be an integer) at so many shekels
per weight, etc. As I tell my students, everything is
discrete or limits of discrete, and for discrete there
is no problem. There IS a problem in passing to the
limit, however, and sets of measure (probability) zero
come up, which are unavoidable.

But we are talking about for beginners - an introduction was requested. I
have no idea whether you are right in what you say, but I was criticising
you for introducing what seems to be far from basic ideas into a discussion
regarding a suitable Introduction to Statistics. I mean that if someone asks
for something basic and in your argument you introduce ideas that by
definition must be beyond the knowledge of the person with the question that
contribution doesn't seem terribly helpful.

Nick


.

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