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.