Optimal robust estimators for families of distributions on the integers

TL;DR

This paper introduces M-estimators with minimal gross error sensitivity for distributions on nonnegative integers, analyzing their properties, asymptotic behavior, and robustness through simulations, especially for Poisson distributions.

Contribution

It derives the M-estimators with the smallest GES for integer-supported distributions and characterizes their asymptotic distributions, including special cases.

Findings

The estimators have asymptotic normal distribution except at certain points.

Simulation results show competitive efficiency and robustness for Poisson distributions.

The estimator's asymptotic behavior varies depending on the distribution's median properties.

Abstract

Let F_{{\theta}} be a family of distributions with support on the set of nonnegative integers 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 (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(F_{n})=umed(F_{{\theta}}), where 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. 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.