Re: Do you want it?
- From: "Greg Heath" <heath@xxxxxxxxxxxxxxxx>
- Date: 17 Aug 2006 00:22:57 -0700
Reef Fish wrote:
Jack Tomsky wrote:
Jack wrote:
*** From the abusive S.S. Wilks, Mathematical
Statistics, equation (8.2.6), page 199, Wiley & Sons,
1962. "Now consider the sample variance, which is
defined as
s^2 = Sum(Xi-Xbar)^2/(n-1)." Jack ***
My response
WHAT IS INEQUIVOCALLY true is that the 2nd moment
about the sample mean is called, BY DEFINITION the
sample variance and that is expression is:
_____= ssd / N
If there are abusers (I never doubt it) they MUST BE
criticized. If we do not do it the confusion grows
up.
Jack, do you really want it?
_____licas (Luis A. Afonso)
I checked every book on my bookshelf. Whenever the population mean is unknown, they all define the sample variance as
s^2 = Sum(Xi-Xbar)^2/(n-1).
In multivariate cases, when the popoulation mean vector is unknown, they all define the sample covariance matrix as
S = Sum(Xi-Xbar)(Xi-Xbar)'/(n-1).
Can you cite a reference which uses your formula?
Jack
To be fair, Afonso did give the Wikipedia reference (though he gave
only
ONE of the two given there:-)) for his "inequivocally true" statement.
http://en.wikipedia.org/wiki/Variance
W> We take a sample of n values from the population, and estimate
W> the variance on the basis of this sample. There are several good
W> estimators. Two of them are well known:
sn^2 ia the one divided by N
and
s^2 is the one divided by (N-1)
W> Both are referred to as sample variance.
I had explain this to Afonso three days ago:
RF> a sample variance is what is computed from a sample, under
RF> some estimation criterion. Whether the denominator is (N-1),
RF> N, or (N+1), they are all called sample variances, for
RF> short, instead of the sample estimate of sigma-squared under t
RF> he criterion of #.
RF> For (N-1), # = Unbiased Estimate.
RF> For (N) # = MLE or maximum likelihood estimate
RF> For (N+1) # = MSE or minimum Mean Squared Error estimate.
The (N+1) is the obscure one. the other two are the ones given in
the Wikipedia webpage.
I think the emphasis in a "sample variance" is that it is obtained from
a SAMPLE of data values. The criterion of estimation does not
alter the fact that they are all called "sample variance", as a short
for of "sample estimate of the population sigma under criterion #".
What modifications to above discussion result when the population
mean, M, is known and the unbiased estimate for the covariance
matrix is
S = Sum(Xi-M)(Xi-M)'/N ?
Hope this helps.
Greg
.
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