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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?

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  • $\begingroup$ Maybe this should be tagged as homework? $\endgroup$
    – whuber
    Commented Nov 11, 2010 at 13:30

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Minimize SSE = Σi (zi-a)2 with respect to a and you get the average of the two sample means, so it looks like your MLE is also your minimum MSE estimator, whether you have one (x,y) pair or several. Not surprising, since the expression for the SSE and the log likelihood function are almost identical.

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