The methods for Smooth estimation of discrete distributions

Abstract

Let X₁,…,X_n be independent and identically distributed random variables with discrete density. This paper gives two methods for smooth estimation of p_i. for each i. One method may be viewed as a generalization to the case of an infinite number of nonzero cell probabilities of the Fienberg and Holland (1970) estimator for the multinomial case. Some Monte Carlo simula-tions show that for a small sample, the proposed estimators yield smaller risks under squared error loss than the usual maximum likelihood estimator (m.l.e.). However, under mild conditions the proposed estimators are shown to have the same large sample properties (strong consistency and asymptotic normality) as the m.l.e.