# Best method to estimate the mean of a normal distribution?

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?

• The trimmed and winsorized mean are pretty popular in robust statistics. – Marc Claesen Mar 2 '14 at 12:55
• $\hat\mu=X_1$ 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. – Scortchi Mar 2 '14 at 15:19
• And note that the Bayesian estimator you give is only asymptotically unbiased - though that's hardly a concern. – Scortchi Mar 2 '14 at 16:09