Re: Corollary: N-P Silliness in Estimation Theory (was: Re: Unusual formulae for confidence intervals)


David Jones wrote:
Reef Fish wrote:
David Jones wrote:
Reef Fish wrote:

But in a sense the MOST "peculiar" of all these concepts is the
N-P theorists' pre-occupation of the notion of an UNBIASED
estimate.
E(statistic) = population parameter to be estimated.

Here, E ( SSE/(n-1)) = sigma^2, hence an unbiased estimate.

but the SQUARE ROOT of S^2 is a BAISED estimate of sigma
whether you use n, (n-1), or (n+1) as the denominator.


On the other hand, scaling by (n-1) is in line with the usual
scaling
in regression by subtracting the number of degrees of freedom
(total
number of regreesors including 1 for the intercept) used in the
model
from the sample size.

But that's only the layman's heuristics without any understanding of
the THEORY of estimation behind all the estimators. In the
regression case, the estimators are all of the UNBIASED class (in
variance), and suffers the same silliness when used for confidence
intervals using the square root for standard error, it's NO LONGER
unbiased!

-- Reef Fish Bob.

But, while "unbiassed-ness" may be the mathematical route by which
form of the basic estimator is derived, the transformed values of the
estimators still retain the property that the transformed value
remains roughly stable as the number of regressors increases.

No. The concept of "unbiasedness" did NOT arise from anything related
to regression. It is simply a theoretical CRITERION of statistical
estimation.

Since the only difference between an unbiased estimate (for the
variance)
and the MLE is (N-1) vs N, any student of mathematics can see that
the different small especially when N is large.

But it's the PRINCIPLE of stressing unbiasedness that is silly.

Your argument would hold water better if it had been stated the OTHER
way, namely, since MLE is invariant under nonlinear transforms, and
is nearly unbiased, why even create, let alone emphasize, the notion
of unbiaseness in the point estimate of an unknown paramete?

That way of thinking would make much more sense than the present
one that one can never be "consistently unbiased" when doing
transformations of parameters.


It seems
possible that this is the sort of property that was being sought and
that the mathematical steps via "unbiassed-ness" is only a convenient
way of acheiving this.

No. It is neither convenient nor logically defensible, especially in
light of the fact that almost everyone uses the BIASED estimate for
an unknown standard deviation, DERIVED from an unbiased
estimate of the variance.

-- Reef Fish Bob.



In this case the scaling at least means that,
for a given fixed sample, the estimate of the error variance
remains
stable as the number of regressors increases, assuming that these
have no explanatory power.

David Jones

.