Cosmological non-Gaussian signature detection: Comparing performance of different statistical tests

Currently, it appears that the best method for non-Gaussianity detection in the cosmic microwave background (CMB) consists in calculating the kurtosis of the wavelet coefficients. We know that wavelet-kurtosis outperforms other methods such as the bispectrum, the genus, ridgelet-kurtosis, and curvelet-kurtosis on an empirical basis, but relatively few studies have compared other transform-based statistics, such as extreme values, or more recent tools such as higher criticism (HC), or proposed "best possible" choices for such statistics. In this paper, we consider two models for transform-domain coefficients: (a) a power-law model, which seems suited to the wavelet coefficients of simulated cosmic strings, and (b) a sparse mixture model, which seems suitable for the curvelet coefficients of filamentary structure. For model (a), if power-law behavior holds with finite 8th moment, excess kurtosis is an asymptotically optimal detector, but if the 8th moment is not finite, a test based on extreme values is asymptotically optimal. For model (b), if the transform coefficients are very sparse, a recent test, higher criticism, is an optimal detector, but if they are dense, kurtosis is an optimal detector. Empirical wavelet coefficients of simulated cosmic strings have power-law character, infinite 8th moment, while curvelet coefficients of the simulated cosmic strings are not very sparse. In all cases, excess kurtosis seems to be an effective test in moderate-resolution imagery.