We consider f-disparities D f (p n ,p n ) between discrete distributions p n =(p n 1 ,...,p n k n ) and their estimates p n =(p n 1 ,...,p n k n ) based on relative frequencies in an i.i.d. sample of size n, where f:(0,~)->R is twice continuously differentiable in a neighborhood of 1 with f''(1)<>0. We derive asymptotic distributions of the disparity statistics nD f (p n ,p n ) under certain assumptions about p n and the second derivatives f'' in a neighborhood of 1. These assumptions are weaker than those known from the literature.