  We systematically analyze the primordial non-Gaussianity estimator used by
the Wilkinson Microwave Anisotropy Probe (WMAP) science team with the basic
ideas of estimation theory in order to see if the limited Cosmic Microwave
Background (CMB) data is being optimally utilized. The WMAP estimator is based
on the implicit assumption that the CMB bispectrum, the harmonic transform of
the three-point correlation function, contains all of the primordial
non-Gaussianity information in a CMB map. We first demonstrate that the
Signal-to-Noise (S/N) of an estimator based on CMB three-point correlation
functions is significantly larger than the S/N of any estimator based on
higher-order correlation functions; justifying our choice to focus on the
three-point correlation function. We then conclude that the estimator based on
the three-point correlation function, which was used by WMAP, is optimal,
meaning it saturates the Cramer-Rao Inequality when the underlying CMB map is
nearly Gaussian. We quantify this restriction by demonstrating that the
suboptimal character of our estimator is proportonal to the square of the
fiducial non-Gaussianity, which is already constrained to be extremely small,
so we can consider the WMAP estimator to be optimal in practice. Our
conclusions do not depend on the form of the primordial bispectrum, only on the
observationally established weak levels of primordial non-Gaussianity.
