Optimal robust estimators for families of distributions on the integers

Abstract

Let Fθ be a family of distributions with support on the set of nonnegative integers Z. 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 (umed(F)) as the median of 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(Fn) = umed(Fθ) , where Fn 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 Fθ(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.