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Science Forum Index » Statistics - Math Forum » Question on Estimator for Variance of Normal...
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| john... |
 Posted: Thu May 08, 2008 6:43 am |
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Guest
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I have a question on the Method of Maximum Likelihood for finding an
Estimator.
The method of MLE when used to find the estimator for the variance of
a Normal Distribution gives the formula which is like the Sample
Variance but instead of dividing by n-1 you divide by n. This
formula is biased ( which is why we use the Sample Variance formula
instead usually ).
But the MLE is. suppose to produce the "minimum variance unbiased
estimator" if one exists... this is what I was told ? So does this
mean that the "minimum variance unbiased estimator" for the variance
of a normal distribution does not exist ?
If it does not exist then why isn't the Sample Variance Formula, which
is unbiased, not the minimum variance unbiased estimator ?
Thanks. |
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| Herman Rubin... |
| Posted: Thu May 08, 2008 8:55 pm |
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In article <045b52ea-44cc-45da-a43c-ea4a976fc8c4 at (no spam) 56g2000hsm.googlegroups.com>,
john <qjohnny2000 at (no spam) gmail.com> wrote:
Quote: I have a question on the Method of Maximum Likelihood for finding an
Estimator.
The method of MLE when used to find the estimator for the variance of
a Normal Distribution gives the formula which is like the Sample
Variance but instead of dividing by n-1 you divide by n. This
formula is biased ( which is why we use the Sample Variance formula
instead usually ).
But the MLE is. suppose to produce the "minimum variance unbiased
estimator" if one exists... this is what I was told ? So does this
mean that the "minimum variance unbiased estimator" for the variance
of a normal distribution does not exist ?
The MLE is not supposed to do that. It does not care if
the variance, or the standard deviation, or the reciprocal
of the variance, is being estimated. These all have
unbiased estimates, at least for n sufficiently large,
and they do not correspond to each other. Which, if any,
should the MLE correspond to?
Quote: If it does not exist then why isn't the Sample Variance Formula, which
is unbiased, not the minimum variance unbiased estimator ?
Thanks.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| ... |
| Posted: Thu May 08, 2008 9:09 pm |
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Guest
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On 8 May 2008 21:55:26 -0400, hrubin at (no spam) odds.stat.purdue.edu (Herman
Rubin) wrote:
Quote: In article <045b52ea-44cc-45da-a43c-ea4a976fc8c4 at (no spam) 56g2000hsm.googlegroups.com>,
john <qjohnny2000 at (no spam) gmail.com> wrote:
I have a question on the Method of Maximum Likelihood for finding an
Estimator.
The method of MLE when used to find the estimator for the variance of
a Normal Distribution gives the formula which is like the Sample
Variance but instead of dividing by n-1 you divide by n. This
formula is biased ( which is why we use the Sample Variance formula
instead usually ).
But the MLE is. suppose to produce the "minimum variance unbiased
estimator" if one exists... this is what I was told ? So does this
mean that the "minimum variance unbiased estimator" for the variance
of a normal distribution does not exist ?
The MLE is not supposed to do that. It does not care if
the variance, or the standard deviation, or the reciprocal
of the variance, is being estimated. These all have
unbiased estimates, at least for n sufficiently large,
and they do not correspond to each other. Which, if any,
should the MLE correspond to?
If it does not exist then why isn't the Sample Variance Formula, which
is unbiased, not the minimum variance unbiased estimator ?
Thanks.
I thought that MLE achieved the Cramer-Rao Lower Bound, and that in a
large sample it is unbiased and minimum variance. Herman, could you
expand on your answer a bit?
Thanks.
-Dick Startz |
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| Herman Rubin... |
| Posted: Sat May 10, 2008 9:19 pm |
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Guest
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In article <7fc724tf4nfjmi2qlbd2r2ilh002tc1oh0 at (no spam) 4ax.com>,
<richardstartz at (no spam) comcast.net> wrote:
Quote: On 8 May 2008 21:55:26 -0400, hrubin at (no spam) odds.stat.purdue.edu (Herman
Rubin) wrote:
In article <045b52ea-44cc-45da-a43c-ea4a976fc8c4 at (no spam) 56g2000hsm.googlegroups.com>,
john <qjohnny2000 at (no spam) gmail.com> wrote:
I have a question on the Method of Maximum Likelihood for finding an
Estimator.
The method of MLE when used to find the estimator for the variance of
a Normal Distribution gives the formula which is like the Sample
Variance but instead of dividing by n-1 you divide by n. This
formula is biased ( which is why we use the Sample Variance formula
instead usually ).
But the MLE is. suppose to produce the "minimum variance unbiased
estimator" if one exists... this is what I was told ? So does this
mean that the "minimum variance unbiased estimator" for the variance
of a normal distribution does not exist ?
The MLE is not supposed to do that. It does not care if
the variance, or the standard deviation, or the reciprocal
of the variance, is being estimated. These all have
unbiased estimates, at least for n sufficiently large,
and they do not correspond to each other. Which, if any,
should the MLE correspond to?
If it does not exist then why isn't the Sample Variance Formula, which
is unbiased, not the minimum variance unbiased estimator ?
Thanks.
Quote: I thought that MLE achieved the Cramer-Rao Lower Bound, and that in a
large sample it is unbiased and minimum variance. Herman, could you
expand on your answer a bit?
Thanks.
-Dick Startz
In an exponential family, the MLE for the observation occurring
in the exponent achieves the Cramer-Rao lower bound; otherwise,
it does not. It need not even have a mean, as the MLE of the
reciprocal of a normal mean, but for iid observations in the
regular case, sqrt(n)*(error in MLE) is asymptotically normal
with mean 0 and variance the Cramer-Rao lower bound. This
latter means that the probability of every closed set approaches
the normal probability of that set, but this does not guarantee
that moments converge.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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