Likelihood Estimator Question?

Show that the maximum likelihood estimator of mu is unbiased while the maximum likelihood estimator of sigma squared is biased, are either of these estimators consistent?|||Is this for a normal distribution?





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Gracious, sorry to take so long! Must be getting forgetful. When you're using the maximum likelihood approach to get estimators for both the mean and variance of a normal distribution, you're right: the estimator for the mean is unbiased while the one for the variance is biased. Since the first is unbiased, then yep, it's consistent. The estimator for the variance, using n samples x1, ..., xn , is





V = (1 / n) ( (x1 - m)^2 + ... + (xn - m)^2 )





where m is the sample mean. If you look closely at E[V], you'll notice that, for example, x1 and m are not quite independent. You'll need to do the legwork, but that's enough to make V biased. That is, E[V] is not exactly the variance of X. But asymptotically, as n gets large, there's little difference. So even though the sample variance is biased, it's consistent. Good luck!