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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](http://en.wikipedia.org/wiki/Estimation_of_covariance_matrices) 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.2012-04-25
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    @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).2012-04-25
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    @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.2012-04-25
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    @leonbloy: I see, thanks, I missed that (about the means being known/unknown).2012-04-25
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    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.2012-04-25
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    thanks for the discussion, it turns out to be biased. I don't see there is an elegant explanation. Anyway, I solved it by work through the double integral. Thanks again2012-04-27
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    You might wish to add at the beginning of your post a mention that (a discussion in the comments showed that) the question asks to prove an invalid result.2012-04-29

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