Hi Yamato,
If M and N are the same, is there any reason to run M simulations and
N simulations separately? What happens if you combine them and
calculate
V and g in the single loop?
I think it gives the wrong answer to do it in a single loop. Note that
the simulation outcomes z are used to compute both V and g, so that
they are quite strongly correlated. In general, if random variables X
and Y are correlated then E[X]E[Y] != E[XY].
A simple example of this problem would be sampling E[X]E[X] for some
random variable X. Let's say X is actually just noise with mean zero.
If you just take one sample x1, then x1*x1 is always +ve, and you
would estimate E[X]E[X]=E[X^2]>0. But if you take two independent
samples x1 and x2, then x1*x2 would be +ve half the time and -ve half
the time, and you would correctly estimate E[X]E[X]=0.
-Dave
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