I'm trying to find an estimator for $\displaystyle \theta.$ I have the following facts:

Where $\displaystyle v_{ij}$ is a matrix that was obtained by multiplying a standard normal matrix and an unspecified data matrix.

$\displaystyle P(sign(v_{1j}) = sign(v_{2j})) = 1 - \frac{\theta}{\pi}$

My problem is how can I estimate $\displaystyle P(sign(v_{1j}) = sign(v_{2j})?$ Is this probability dependent on the distribution of $\displaystyle v_{ij}?$ Or, is the desired result simply:

$\displaystyle P(sign(v1)=sign(v2))= 1/3$, since,

+ & -, - & +, 0 & +,
+ & 0, - & 0, 0 & -
+ & +, - & -, 0 & 0,

And 3/9 satisfy the condition.

Thanks