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I met the following problem when doing estimation and detection homework. The problem asks for a maximum likelihood estimator for (v,$\rho$) of bivariate joint Gaussian, where v is the common variance for Y1 and Y2, rho is correlation coefficient, Y1, Y2 mean zero. Given one pair of observation (y1, y2).

I find that the MLE for $\rho$ is $\hat{\rho}=\frac{2y_1y_2}{y_1^2+y_2^2}$, and $\hat{v}=\frac{y_1^2+y_2^2}{2}$. now I need to show that these estimators are unbiased. The unbiasedness of $\hat{v}$ is easy. Then I need to verify $$E[\hat{\rho}]=\rho$$

I managed to show this by working through the double integral and change to polar coordinate, but the computation is heavy.

Here is my question, is there a easier (neat) solution to show it is unbiased?

Thanks in advance.

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This might be helpful. It says that the maximum likelihood estimator is biased and you get an unbiased estimator by replacing $n$ by $n-1$, though. –  joriki Apr 25 '12 at 3:01
    
@joriki: the problem is that, the (un)biasness of some estimator of the covariance is not trivially related to the (un)biasness of the estimator of the correlation coefficient (obtained by dividing the previous one by the estimator of the common variance). –  leonbloy Apr 25 '12 at 11:13
    
@joriki: Further, diving by $n$ gives a biased estimator of the covariance when the mean is unknown. When the mean is know (as here) it's unbiased. –  leonbloy Apr 25 '12 at 11:20
    
@leonbloy: I see, thanks, I missed that (about the means being known/unknown). –  joriki Apr 25 '12 at 11:22
    
It's easy to see that $y_1 y_2$ is the MLE-unbiased estimator of $Cov(y_1 y_2)$ and $(y_1^2 +y_2^2)/2$ the MLE-unbiased estimator of $\sigma^2$ and, $\rho = Cov(y_1 y_2)/\sigma^2$. But the ratio of unbiased estimators is not necessarily unbiased. I wonder if there is some elegant reason why that's true in this case. –  leonbloy Apr 25 '12 at 17:50

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