The CMB Bispectrum, Trispectrum, non-Gaussianity, and the Cramer-Rao
  Bound
  Minimum-variance estimators for the parameter fnl that quantifies local-model
non-Gaussianity can be constructed from the cosmic microwave background (CMB)
bispectrum (three-point function) and also from the trispectrum (four-point
function). Some have suggested that a comparison between the estimates for the
values of fnl from the bispectrum and trispectrum allow a consistency test for
the model. But others argue that the saturation of the Cramer-Rao bound by the
bispectrum estimator implies that no further information on fnl can be obtained
from the trispectrum. Here we elaborate the nature of the correlation between
the bispectrum and trispectrum estimators for fnl. We show that the two
estimators become statistically independent in the limit of large number of CMB
pixels and thus that the trispectrum estimator does indeed provide additional
information on fnl beyond that obtained from the bispectrum. We explain how
this conclusion is consistent with the Cramer-Rao bound. Our discussion of the
Cramer-Rao bound may be of interest to those doing Fisher-matrix
parameter-estimation forecasts or data analysis in other areas of physics as
well.
