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

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

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

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

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

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

And 3/9 satisfy the condition.

Thanks