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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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    $\begingroup$ Why don't you post here: stats.stackexchange.com This is a statistics question. $\endgroup$
    – Richi Wa
    Mar 12, 2015 at 15:43
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    $\begingroup$ The MLE variance estimator for normal distributions is biased because it divides by $n$ rather than $n-1$, see ee.columbia.edu/~dliang/files/mle_biased.pdf. Not sure how much that relates to the above quote. $\endgroup$
    – John
    Mar 12, 2015 at 18:15
  • $\begingroup$ You argument is irrelevant to my question. The statement is that if there is a sufficient statistic for an exponential distribution that would be MLE. $\endgroup$ Mar 13, 2015 at 17:59
  • $\begingroup$ In fact this sentence "which means that when minimum variance unbiased estimators exist, they will be MLEs." is what does not make sense. $\endgroup$ Mar 13, 2015 at 18:03
  • $\begingroup$ Richard - I agree to some sense. However, I see people using MLE a lot in practice, but the statement indicates that the estimates are not close enough for small samples and specially for non-expoential distributions. $\endgroup$ Mar 13, 2015 at 18:08

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