Preliminary Phi-divergence test estimator for multinomial probabilities

The four minimum Phi-divegence estimators of multinomial probabilities under a model constraint, namely, unrestricted estimator, restricted estimator, preliminary test estimator, shrinkage estimator, and the positive-rule shrinkage estimator, have been studied. The asymptotic distribution of these estimators are given under Pittman alternatives, together with the asymptotic bias and the asymptotic risk under the quadratic loss. Comparisons within these four estimators using their asymptotic function risks are made, and several conclusions are obtained