Re: [OT][Long][You All Know Everything Dept] Recommendations for Stat book?

From: R. Scott McIntire (mcintire_charlestown_at_comcast.net)
Date: 06/11/04 

Date: Fri, 11 Jun 2004 02:55:00 GMT

"Kenny Tilton" <ktilton@nyc.rr.com> wrote in message
news:YK%xc.172576$WA4.15245@twister.nyc.rr.com...
>
>
http://www.amazon.com/exec/obidos/tg/detail/-/0534357784/qid=1086880204/sr=1-3/ref=sr_1_3/002-3751143-0849664?v=glance&s=books
>
> I found it to be too light in terms of content. For example, it defined
> in 2nd chapter what
> population variance and sample variance is, and that in case of one you
> divide by n and in case of
> the other by n-1. Later in the text, and I quote:
>
> "You may wonder why you need to divide by (n-1) rather than n when
> computing sample variance ...
> turns out that the sample variance s^2 with (n-1) in the denominator
> provides better estimates of
> (sigma)^2 than would an estimator calculated with n in the denominator"
>
> And that's it. As a mathematically curious person with college calculus
> experience, I find that to
> be extremely intellectually unsatisfying.

I usually find that you get books that a recipies or a theorem definition
parade, when what you would like is some intuition about the subject backed
up with a reaonable explaination. Let me explain a little bit about sample
variance.

It turns out that if I you n identically distributed random variables x_i
{i=1,..n} with mean m and standard variance s^2, then people combine these
in order to produce an estimator for the mean or the standard deviation. If
we take the n random variables and form a NEW RANDOM VARIABLE x_bar that is
the average value of the other random variables then this new random
variable can be used as an estimator of the mean m. It is also an UNBIASED
estimator as the expected value of this random variable is m. You can do the
same thing to get an estimator of the variance s^2. The random variable
s_bar^2 = (sum (x_i - m)^2 )/ n. You can check that the expectation of this
random variable is s^2.

is an unbiased estimator of the variance of the random variables. That is,
the expected value of this random variable is s^2. Now, although we know
that
all the random variables have the same mean, m, and variance, s^2, suppose
we don't know what they are. Well we didn't need to know them for the first
estimator - for the mean - ok. But, for the variance estimator, we need to
know m. If we don't have it, we could try to use the estimator for m. This
gives a new estimator for the variance:
s_2^2 = (sum x_i - x_bar)^2 / n.

But now this estimator is unbiased. The expectation of this random variable
is not s^2. A tedious calculation will show that you need to divide by n-1
rather than n to make it unbiased. Finally, in the real world we replace the
x_i's with actual sample data that is suppose to come from identically
distributed random variables.

Hope this helps.

-R. Scott McIntire

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