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Shouldnt we also note that the MLE is biased, and include a section on an unbiased estimator?
Michael, please see if my piece fits well enough. -- Stas Kolenikov
I'd like to point out that the symbol starts out denoting a parameter, but then is later used as an estimator. Btyner 19:40, 9 February 2007 (UTC)
I have rewritten the summary, putting the maximum likelihood estimator in perspective.
The use X and x should be made consistent throughout the article.
I am not sure about the purpose of the statement
Although no one is surprised that the estimator of the population covariance matrix is [closely related to] the sample covariance matrix, the mathematical derivation is perhaps not widely known and is surprisingly subtle and elegant.
for the following reasons:
Jmath666 14:34, 25 March 2007 (UTC)
I have now removed the statement above as proposed. Jmath666 23:05, 31 March 2007 (UTC)
In the beginning of the article, where we explaining way we divide by the factor n-1 rather than with n (in the unbiased estimator for the cov matrix), it is said that the reason is because the mean is not known and is replaced by the sample mean. Of course this is true. And more then that, it directs us to the notion of degrees of freedom. and thus should link to the article on Degrees of freedom (statistics). What do you think ? Talgalili 13:04, 25 August 2007 (UTC)
quote: Given a sample consisting of independent observations X1,..., Xn of a random vector X ∈ Rp×1 (a p×1 column),
It has been said in wikipedia:Village_pump_(proposals)#Wait!!! that this should be clarified to the benefit of nonexpert readers. I too find it obscure. Is X1 an observation of a random vector X with p components, or is X1 an observation of the first composant of a random vector X with n components? Is p=n? If not, what is p? Conventionally, (X1,..., Xn) are the components of the n-vector X. That is apparently not the case here. Please clarify. Bo Jacoby ( talk) 08:26, 9 August 2008 (UTC).
Thank you, Pdbayley, for your edit. However the typo, if it was a typo? is also present in the very first line of the article. Perhaps the covariance matrix is a p×p matrix, and each and are p-vectors? I now tend to think that this is the case. Probably the observations should be written x rather than X.
Bo Jacoby ( talk) 08:14, 11 August 2008 (UTC).
Given a sample consisting of independent observations X1,..., Xn of a random vector so that, for each vector, Xi ∈ Rp×1 (a p×1 column), an unbiased estimator of the (p×p) covariance matrix
is the sample covariance matrix
where the vector is given by
I was startled at how abrupt and devoid of context-setting the opening sentence is:
I wondered if I had written that. I looked at the history. I did not write that. I knew that some people start Wikipedia articles like this; I hadn't realized that some people alter Wikipedia articles with proper introductory material so that they read like that. Michael Hardy ( talk) 13:03, 11 August 2008 (UTC)
I have removed the restored para, as the article is not immediately about any of
The point made was already covered later in the article, but might be given a different placement if emphasis is required. I have added a more general intro and tried to separate-off the bits related to the normal distribution by putting them into a separate section. Melcombe ( talk) 15:04, 11 August 2008 (UTC)
What is known about the general, non-gaussian case? In what sense does the sample covariance estimate/converges to the covariance matrix then? What if the random vectors are not independent but only approximately so? Jmath666 ( talk) 05:45, 9 November 2008 (UTC)
I have some issues with the new section recently added. I've commented it out in the article for now.
The first-order conditions for a MLE of parameter θ are that the first derivative of the log-likelihood function should be null at θMLE. Intuitively, the second derivative of the log-likelihood function indicates its curvature : the higher it is, the better identified θMLE since the likelihood function will be inverse-V-shaped around θMLE. Formally, it can be proved that
where can be estimated by
In this case "general case" appears not to mention covariance matrices at all, but as applied to covariance matrices being estimated, it seems to say that the sample covariance matrix has approximately a normal distribution with a variance Ω. I'm not sure I know what is meant by the variance of a matrix-valued random variable. I may have seen it defined at some point in the past, but I think if one writes about such a thing here, the article should explain what it is. And the parameter θ would in this case be a positive-definite matrix. How do you compute those partial derivatives with respect to a positive-definite matrix-valued random variable? I don't know for sure whether I've seen such a thing, but I think if that is to be done here, the article should explain what it is.
Maybe the new paragraph belongs in the main maximum likelihood article or one of the other related articles. Michael Hardy ( talk) 20:23, 6 May 2009 (UTC)
Donald Andrews and others have written on median-unbiased estimators, with application to time series analysis. Kiefer.Wolfowitz ( Discussion) 14:59, 13 April 2011 (UTC)
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Shouldnt we also note that the MLE is biased, and include a section on an unbiased estimator?
Michael, please see if my piece fits well enough. -- Stas Kolenikov
I'd like to point out that the symbol starts out denoting a parameter, but then is later used as an estimator. Btyner 19:40, 9 February 2007 (UTC)
I have rewritten the summary, putting the maximum likelihood estimator in perspective.
The use X and x should be made consistent throughout the article.
I am not sure about the purpose of the statement
Although no one is surprised that the estimator of the population covariance matrix is [closely related to] the sample covariance matrix, the mathematical derivation is perhaps not widely known and is surprisingly subtle and elegant.
for the following reasons:
Jmath666 14:34, 25 March 2007 (UTC)
I have now removed the statement above as proposed. Jmath666 23:05, 31 March 2007 (UTC)
In the beginning of the article, where we explaining way we divide by the factor n-1 rather than with n (in the unbiased estimator for the cov matrix), it is said that the reason is because the mean is not known and is replaced by the sample mean. Of course this is true. And more then that, it directs us to the notion of degrees of freedom. and thus should link to the article on Degrees of freedom (statistics). What do you think ? Talgalili 13:04, 25 August 2007 (UTC)
quote: Given a sample consisting of independent observations X1,..., Xn of a random vector X ∈ Rp×1 (a p×1 column),
It has been said in wikipedia:Village_pump_(proposals)#Wait!!! that this should be clarified to the benefit of nonexpert readers. I too find it obscure. Is X1 an observation of a random vector X with p components, or is X1 an observation of the first composant of a random vector X with n components? Is p=n? If not, what is p? Conventionally, (X1,..., Xn) are the components of the n-vector X. That is apparently not the case here. Please clarify. Bo Jacoby ( talk) 08:26, 9 August 2008 (UTC).
Thank you, Pdbayley, for your edit. However the typo, if it was a typo? is also present in the very first line of the article. Perhaps the covariance matrix is a p×p matrix, and each and are p-vectors? I now tend to think that this is the case. Probably the observations should be written x rather than X.
Bo Jacoby ( talk) 08:14, 11 August 2008 (UTC).
Given a sample consisting of independent observations X1,..., Xn of a random vector so that, for each vector, Xi ∈ Rp×1 (a p×1 column), an unbiased estimator of the (p×p) covariance matrix
is the sample covariance matrix
where the vector is given by
I was startled at how abrupt and devoid of context-setting the opening sentence is:
I wondered if I had written that. I looked at the history. I did not write that. I knew that some people start Wikipedia articles like this; I hadn't realized that some people alter Wikipedia articles with proper introductory material so that they read like that. Michael Hardy ( talk) 13:03, 11 August 2008 (UTC)
I have removed the restored para, as the article is not immediately about any of
The point made was already covered later in the article, but might be given a different placement if emphasis is required. I have added a more general intro and tried to separate-off the bits related to the normal distribution by putting them into a separate section. Melcombe ( talk) 15:04, 11 August 2008 (UTC)
What is known about the general, non-gaussian case? In what sense does the sample covariance estimate/converges to the covariance matrix then? What if the random vectors are not independent but only approximately so? Jmath666 ( talk) 05:45, 9 November 2008 (UTC)
I have some issues with the new section recently added. I've commented it out in the article for now.
The first-order conditions for a MLE of parameter θ are that the first derivative of the log-likelihood function should be null at θMLE. Intuitively, the second derivative of the log-likelihood function indicates its curvature : the higher it is, the better identified θMLE since the likelihood function will be inverse-V-shaped around θMLE. Formally, it can be proved that
where can be estimated by
In this case "general case" appears not to mention covariance matrices at all, but as applied to covariance matrices being estimated, it seems to say that the sample covariance matrix has approximately a normal distribution with a variance Ω. I'm not sure I know what is meant by the variance of a matrix-valued random variable. I may have seen it defined at some point in the past, but I think if one writes about such a thing here, the article should explain what it is. And the parameter θ would in this case be a positive-definite matrix. How do you compute those partial derivatives with respect to a positive-definite matrix-valued random variable? I don't know for sure whether I've seen such a thing, but I think if that is to be done here, the article should explain what it is.
Maybe the new paragraph belongs in the main maximum likelihood article or one of the other related articles. Michael Hardy ( talk) 20:23, 6 May 2009 (UTC)
Donald Andrews and others have written on median-unbiased estimators, with application to time series analysis. Kiefer.Wolfowitz ( Discussion) 14:59, 13 April 2011 (UTC)