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.

中文翻译：

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整数分布族的最优鲁棒估计量

令 $$F_{\theta }$$ F θ 是支持非负整数集 $$Z_{0}$$ Z 0 的分布族。在本文中，我们推导出具有最小总误差敏感度 (GES) 的 M 估计量。我们首先将支持 $$Z_{0}$$ Z 0 (umed( F )) 的分布 F 的统一中位数定义为 $$x+u,$$ x + u 的中位数，其中 x 和 u是独立变量，分别在 [-0.5,0.5] 中具有分布 F 和均匀分布。在一些一般条件下，我们证明具有最小GES的估计量满足umed $$(F_{n})=$$ ( F n ) = umed $$(F_{\theta }),$$ ( F θ ) ，其中$ $F_{n}$$ F n 是经验分布。找到这些估计量的渐近分布。这种分布是正态分布，除非存在正整数 k 使得 $$F_{\theta }(k)=0.5.$$ F θ ( k ) = 0.5 。在最后一种情况下，渐近分布在 0 的每一侧表现正常，但具有不同的方差。模拟 Monte Carlo 研究针对泊松分布，将该估计器的有限样本量的效率和稳健性与其他稳健估计器的效率和稳健性进行了比较。