Optimal robust estimators for families of distributions on the integers

Let $$F_{\theta }$$ F θ be a family of distributions with support on the set of nonnegative integers $$Z_{0}$$ Z 0 . In this paper we derive the M-estimators with smallest gross error sensitivity (GES). We start by defining the uniform median of a distribution F with support on $$Z_{0}$$ Z 0 (umed(F)) as the median of $$x+u,$$ x + u , where x and u are independent variables with distributions F and uniform in [-0.5,0.5] respectively. Under some general conditions we prove that the estimator with smallest GES satisfies umed $$(F_{n})=$$ ( F n ) = umed $$(F_{\theta }),$$ ( F θ ) , where $$F_{n}$$ F n is the empirical distribution. The asymptotic distribution of these estimators is found. This distribution is normal except when there is a positive integer k so that $$F_{\theta }(k)=0.5.$$ F θ ( k ) = 0.5 . In this last case, the asymptotic distribution behaves as normal at each side of 0, but with different variances. A simulation Monte Carlo study compares, for the Poisson distribution, the efficiency and robustness for finite sample sizes of this estimator with those of other robust estimators.