# What is the minimum mean squared error estimator of the mean of two normally-distributed variables?

Let $Z=(X+Y)/2$, where $X$ and $Y$ are independent normally-distributed random variables with known variances $\sigma^2_X$ and $\sigma^2_Y$ and unknown (and possibly different) means. Given a sample $x_1$ from $X$ and $y_1$ from $Y$, what is the minimum mean squared error estimator of the mean of $Z$? Is there a biased estimator with an MSE improved over the maximum likelihood estimator $\frac{x_1+y_1}{2}$? Can we generalize to $n$ mutually-independent random variables?

• Maybe this should be tagged as homework? – whuber Nov 11 '10 at 13:30