Optimal robust for families of distributions on the integers

MARONNA, RICARDO; YOHAI, VICTOR JAIME

STATISTICAL PAPERS

SPRINGER

Lugar: Berlin; Año: 2021 vol. 62 p. 2269 - 2281

0932-5026

Let Fθ be a family of distributions with support on the set of nonnegative integers Z0. 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 Z0 (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.