  Since the inhomogeneous instrument noise can produce extra non-Gaussianity in
the CMB anisotropy, its effect should be carefully subtracted in the primordial
non-Gaussianity estimation. We calculate the probability distribution function
of the CMB anisotropy for local type of non-Gaussianity, from which the optimal
estimator in the general case (inhomogeneous noise and cut sky) is obtained.
The new estimator obtained here is different from the popular one, since the
inhomogeneous noise and cut sky effects are completely accounted. The CMB
anisotropy in the new estimator is noise weighted. The noise weight is
different from that used by WMAP Group in their 5-year data analysis. Although
it is still difficult to calculate the new estimator rigorously, for the case
of the slightly inhomogeneous noise, there exists a series expansion method to
compute the new estimator. Each order in the series is suppressed by two
factors, $(\frac{\sigma^2}{\sigma^2_{i}}-1)$ and $\frac{C_l}{C^{tot}_{l}}$,
which make the method feasible. Through the Edgeworth expansion we can
generalize our discussion to other types of non-Gaussianity.
