Let $X = ( x_1, ..., x_n ) $ be $n$ samples from a normal distribution with unknown mean. What is the best estimator for this mean?

I can think of at least 2 unbiased estimators:

  • The empirical mean $ \hat \mu_1 = \frac{\sum_i x_i }{n} $
  • A bayesian approach $ \hat \mu_2 = E[ P( \theta | X ) ] $, where $ P( \theta | X ) $ is the posterior distribution over the mean. The prior distribution over the mean is $ \cal N ( m, s^2) $ for some meta parameters $(m, s^2)$.

Are there other biased or unbiased estimators for this problem? Which one is the best?

  • 3
    $\begingroup$ The trimmed and winsorized mean are pretty popular in robust statistics. $\endgroup$ – Marc Claesen Mar 2 '14 at 12:55
  • 5
    $\begingroup$ $\hat\mu=X_1$ is an unbiased estimator. The estimator $\hat\mu=7$, though biased, can perform better than the sample mean under some circumstances. Nevertheless the sample mean has the advantage of being the uniformly minimum-variance unbiased estimator & the maximum-likelihood estimator of the mean for a normal population, which is why alternatives are only used in practice when the normality assumption is doubted. $\endgroup$ – Scortchi Mar 2 '14 at 15:19
  • 5
    $\begingroup$ And note that the Bayesian estimator you give is only asymptotically unbiased - though that's hardly a concern. $\endgroup$ – Scortchi Mar 2 '14 at 16:09

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