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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.

Optimal robust estimators for families of distributions on the integers