AbstractSuppose the multinomial parameterspr(θ) are functions of a real valued parameter 0,r= 1,2, …,k.A minimum discrepancy (m.d.) estimator θ of θ is defined as one which minimises the discrepancy functionD= Σ kr = 1nrf(pr/nr), for a suitable functionfwherenris the relative frequency inr‐th cell,r=1,2, …, k.All the usual estimators like maximum likelihood (m. l), minimum chi‐square (m. c. s.)., etc. arem.d.estimators. Allm.d.estimators have the same asymptotic (first order) efficiency. They are compared on the basis of their deficiencies, a concept recently introduced by Hodges and Lehmann [2]. The expression for least deficiency at any θ is derived. It is shown that in general uniformly least deficient estimators do not exist. Necessary and sufficient conditions onpr(0) form. t.andm. c. s.estimators to be uniformly least deficien