Why the Normal MLE Variance Estimate Is Downward Biased

Summary

The document raises a question about why the maximum-likelihood estimate of variance under a normal model is downward biased, despite a cited general claim about the optimality of maximum-likelihood estimators in exponential families. It frames a useful distinction between finite-sample unbiasedness and the large-sample properties often emphasized for maximum likelihood.

However, the text contains only the question and a quotation from an econometrics discussion; it provides no derivation, answer, or numerical evidence. It therefore identifies a statistical issue rather than teaching a complete solution. Any reading note should treat the proposed contradiction as unresolved here: the document does not explain the role of the estimated mean, specify the variance estimator being compared, or discuss conditions under which an unbiased estimator exists.

Key ideas

• The document asks why the normal-model maximum-likelihood variance estimator is downward biased.
• It contrasts a general claim about exponential-family maximum-likelihood estimators with finite-sample bias.
• No derivation or answer is provided, so the issue remains unresolved in the source.

Tags

# MLE estimate of normal distribution


# MLE estimate of normal distribution












Probably a naive question. I am quoting this from Greene's econometrics book:

"The occasional statement that the properties of the MLE are only optimal in large samples is not true, however. It can be shown that when sampling is from an exponential family of distributions, there will exist sufficient statistics. If so, MLEs will be functions of them, which means that when minimum variance unbiased estimators exist, they will be MLEs. [See Stuart and Ord (1989).] Most applications in econometrics do not involve exponential families, so the appeal of the MLE remains primarily its asymptotic properties."

So why the variance estimate is downward biased in MLE for a normal distribution?

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