CMB bispectrum, trispectrum, non-Gaussianity, and the Cramer-Rao bound
Minimum-variance estimators for the parameter f_(nl) 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 f_(nl) from the bispectrum and trispectrum allow a consistency test for the model. But others argue that the saturation of the Cramer-Rao bound—which gives a lower limit to the variance of an estimator—by the bispectrum estimator implies that no further information on f_(nl) can be obtained from the trispectrum. Here, we elaborate the nature of the correlation between the bispectrum and trispectrum estimators for f_(nl). 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 f_(nl) 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.
Additional Information© 2011 American Physical Society. Received 5 October 2010; published 14 January 2011. M. K. thanks the support of the Miller Institute for Basic Research in Science and the hospitality of the Department of Physics at the University of California, Berkeley, where part of this work was completed. M. K. was supported at Caltech by DoE Grant No. DE-FG03-92-ER40701, NASA Grant No. NNX10AD04G, and the Gordon and Betty Moore Foundation.
Published - Kamionkowski2011p12686Phys_Rev_D.pdf