On 6/30/07, John Randall <[EMAIL PROTECTED]> wrote:
The above summarizes the estimation process I am doing. I now believe
you are doing something different, which explains why we are having
trouble communicating.
You have a population with known distribution f, and a random variable
X whose distribution is f. You then construct a sample space of
equiprobable outcomes and define a random variable Y on this whose
distribution is g, with g is an approximation to f. Then then mean
and variance of Y, are expected to approximate the mean and variance
of X.
For you, the sample mean is E(Y), a number approximating E(X), and the
sample variance is \sigma^2(Y), a number approximating \sigma^2(X).
This explains some of the confusion we have had, where I have been
insisting that the sample mean is a random variable, and you have been
insisting it is a number.
The term "sample space" is misleading. For the finite distributions
we are discussing, the sample space is just the set of outcomes.
Applying a random variable to this is a sample of size 1: if S is the
sample space, X is a function X:S->R with distribution f.
When I talk about a sample of size n, I am talking about a function
X1 x X2 x X3 x...Xn : S x S x ... x S-> R x R x ... x R,
with distribution f x f x ... x f. This random variable ranges over
all possible samples of size n.
Please let me know if this accurately represents what you are doing.
First off, I completely agree with your characterization of my difficulties
as communication difficulties.
As I currently understand things, I would say that I am struggling with
ambiguities in terminology -- where things that I think of as critical are
typically ignored -- I presume because other people think of them as
uninteresting or distracting.
That said... I want to be able to talk about populations, distributions,
samples, etc. using a consistent set of terminology regardless of
whether or not the distribution of the population is known or unknown,
or partially understood.
So while your above description does characterize one aspect of what
I want to be able to treat, I also want to be able to treat the cases that
you are interested in (which you had described earlier in your message,
in the text I elided).
As I currently understand things, this means that I need to work with
two different mechanisms for computing variance / standard deviation
when talking about small finite populations and samples (one with the
unbiased estimator and, one with no estimator) and I also need to find
some way of talking about samples and populations at different levels
of abstraction (examples vs. more general cases).
In recent messages, I have started describing standard deviation with
no estimator as "population standard deviation" and standard deviation
with the unbiased estimator as "sample standard deviation", and I
am reasonably comfortable with that nomenclature. But I think that
this conflicts with your description of how you proceed . I guess I can
live with that, as this distinction seems irrelevant with infinite
populations, but I am still grappling with the way this issue gets
treated in some contexts.
Thanks,
--
Raul
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