I am re-stating this question for my own intuition, and for any useful
feedback it may generate...
Can the value of any variable ever be "infinity"? I didn't think infinity
was a number (or is it?). Assuming you have one realization (sample) of a
finite normal population, then any particular value drawn from this sample
could still be "as large as it wants to be" unless you define the mean and
variance, and it may still (with very small probability) be an extremely
large value...
If /\ is a normal distribution centered at the intersection of "/" and "\",
then, this "finite population" can be located anywhere along the following
line...
/\ ... /\ ...
/\ ...
-inf
<---------------------------------------------------------------------------
--------> inf
So, in reference to Peter's post, how does the idea that it is a "finite"
population (/\) have anything to do with the value of the maximum being
extremely ("infinitely") large (being more 'left' on the above line)? I
guess it might be because the "finite" sample is a "biased" version of the
"infinite population" since you would not expect to see "infinitely large"
values in the "finite sample"?
hmmm, maybe I answered my own question... or not.
p
----- Original Message -----
From: "Peter Flom" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Sent: Monday, March 22, 2004 10:37 AM
Subject: Re: [edstat] Point Estimator for the Maximum
> Clearly Don is correct that it depends on the nature of the distribution
> of the variable in a particular population (and on population size).
>
> But given a finite population, I don't think the expected value of the
> maximum of a normally distributed variable is infinity. (But maybe I am
> all wrong....)
>
> Given a population size N and a distribution, could one not bootstrap
> for the maximum? I know that in the typical case, where one wants to
> estimate the population parameter from a sample statiistic, the maximum
> is not bootstrappable (if that's the word) but here, if we are given the
> population size and the distribution, it seems to me intuitively that it
> should work
>
> e.g. If we are willing to posit that IQ is normally distributed with
> mean 100 and sd 15 (there is evidence that this is not the case, but
> let's ignore that for this example) and we wanted to estimate the
> highest IQ in the USA (let's say the population is 220 million), why
> couldn't we repeatedly find the maximum of a sample of 220 million,
> distributed this way?
>
> As a quick shot, I ran this in R
>
> max(rnorm(220000000, 100, 15))
>
> three times and got values of 180.23, 181.40, and 180.56.
>
> I can't prove that this is correct, but it seems sensible, and the
> results also make sense
>
> Or am I doing something all wrong here?
>
> Peter
>
>
>
> >>> [EMAIL PROTECTED] 3/22/2004 11:02:02 AM >>>
> Surely this depends heavily on "the population"; which I think must
> be
> finite, or at any rate bounded, for the question even to make sense.
> (What's the population maximum of a normal distribution? Infinity, I
> should think -- which is unlikely to be well estimated by any sample
> value.) Perhaps there were some (implied?) conditions, or
> constraints,
> in the context of the original question?
> -- Don.
>
> On Fri, 19 Mar 2004, Kaplon, Howard wrote:
>
> > A student asked a colleague of mine if the sample maximum was the
> best
> > point estimator of the population maximum. While I was tempted to
> say
> > yes, the idea that this must surely be a biased under-estimator came
> > to mind. Can someone answer the problem and point me to a source?
>
> ------------------------------------------------------------
> Donald F. Burrill [EMAIL PROTECTED]
> 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
> .
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