Deficiencies of minimum discrepancy estimators

Suppose the multinomial parameters p r (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 function D = Σ   k r = 1n r f(p r /n r ), for a suitable function f where n r is the relative frequency in r ‐th cell, r = 1,2, …, k. All the usual estimators like maximum likelihood ( m. l ), minimum chi‐square ( m. c. s. )., etc. are m.d. estimators. All m.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 on p r (0) for m. t. and m. c. s. estimators to be uniformly least deficient are obtained.