Higher Criticism statistic: detecting and identifying non-Gaussianity in the WMAP first-year data

Abstract

Higher criticism is a recently developed statistic for the detection of non-Gaussianity. It was proposed by Donoho & Jin, who showed it to be effective at resolving a very subtle testing problem: whether n normal means are all zero versus the alternative that a small fraction is non-zero. Higher Criticism is also useful in the detection of the non-Gaussian convolution component of cosmic strings in the cosmic microwave background (CMB) (see Jin et al.). In this paper, we study how well the anisotropies of the CMB fit with the homogeneous and isotropic Gaussian distribution predicted by the standard inflationary model. We find that Higher Criticism is useful for two purposes. First, Higher Criticism has competitive detection power, and non-Gaussianity is detected at the level of 99 per cent in the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data. We generated 5000 Monte Carlo Gaussian simulations of the CMB maps. By applying the Higher Criticism statistic to all of these maps in wavelet space, we constructed confidence regions of Higher Criticism at levels of 68, 95 and 99 per cent. We find that the Higher Criticism value of the WMAP data is outside the 99 per cent confidence region at a wavelet scale of 5 ◦ (99.46 per cent of Higher Criticism values based on simulated maps are below the values for WMAP). Secondly, Higher Criticism offers a way to locate a small portion of data that accounts for the non-Gaussianity. This property is not immediately available for other statistical tests such as the widely used excess kurtosis test. Using Higher Criticism, we have successfully identified a ring of pixels centred at l ≈ 209 ◦ , b ≈− 57 ◦ , which seems to account for the observed detection of non-Gaussianity at the wavelet scale of 5 ◦ . After removal of the ring from the WMAP data set, no more prominent deviation from Gaussianity was found. Note that the detection was achieved in wavelet space first. Secondly, it is always possible that a fraction of pixels within the ring might deviate from Gaussianity even if they do not appear to be above the 99 per cent confidence level in wavelet space. The location of the ring coincides with the cold spot detected by Vielva et al. and Cruz et al.