Nonclassicality criteria based on intensity moments and derived from the usual matrix approach are compared to those provided by the majorization theory. The majorization theory is shown to give a greater number of more suitable nonclassicality criteria. Fifteen experimentally useful criteria of the majorization theory containing the intensity moments up to the fifth order are identified. Their performance is experimentally demonstrated on the set of eleven potentially nonclassical states generated from a twin beam by postselection based on detecting a given number of photocounts in one arm by using an iCCD camera. ; Comment: 14 pages, 4 figures