  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 8-th moment, excess kurtosis is an asymptotically
optimal detector, but if the 8-th 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
8-th 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.
