An extended Kolmogorov phenomenology for the dimensional analysis of t
he energy spectrum has been introduced by Kraichnan [Phys. Fluids 8 (1
965) 1385]. Kraichnan's ideas are here extended to show how the higher
-order spectral moments can scale as k-5/3, for any order and without
the assumption of a Gaussian process. New experimental data on higher-
order spectra is presented that exhibits this k-5/3 scaling.