Recent research on texture synthesis suggests that characterisation of those properties of textures to which human observers are sensitive may be provided by the histograms of the coefficients of a wavelet decomposition. In this study we examined the properties of wavelet histograms that affect texture discrimination by measuring observer sensitivity to differences in the wavelet histograms of synthetic textures. The textures, generated via Gabor micropattern synthesis, were broadband, with amplitude spectra that are characteristic of natural images, i.e. 1/f. We measured texture-difference thresholds for three moments of the wavelet histograms -- variance, skew and kurtosis -- by manipulating the contrast, phase, and density, of the Gabor elements used to construct the textures. Observers discriminated more efficiently between textures that had differences in kurtosis, than between textures that had differences in either variance or skew. Performance was compared to two model observers; one used the pixel-luminance histogram, the other used the histogram of the output of wavelet-filters. The results support the idea that the visual system is relatively sensitive to the kurtosis, or 4th moment, of the wavelet histogram of textures. We argue that higher than 4th-order moments will, in practice, become increasingly difficult for the visual system to represent because the lack of a perfect match between the elements and the receptive fields effectively blurs the response histogram, thereby attenuating higher moments.