Distribution of estimators from Gaussian simulations (L_max = 10) compared to data estimates (vertical lines). The distribution is rather skewed because the main contributions are from L ≲ 4 where the modulation power spectra have skewed χ² distributions with low degrees of freedom. The red line shows the predicted distribution for a weighted sum of estimators assuming the reconstructed modulation modes are Gaussian with 2L + 1 modes measured per L, which fits the full simulations well. The black vertical lines show the data estimates from L_max = 10, and should be compared to the green which have L_max = 2 and hence are insensitive to the anomalous octopole signal. The dashed lines are τ_NL,1, the slightly more optimal variant of the estimator.