Consistency of deviance-based M estimators

Abstract

In a general estimation problem, the deviance function generates statistics that are similar to squared standardized residuals. A deviance-based M estimator (DBME) is defined as an adaptively weighted maximum-likelihood estimator, where the weights depend upon the deviances. In both a single-parameter and a regression setting, we give some general conditions under which a DMBE is consistent. For a suitable weighting scheme, these conditions are satisfied in many continuous Cramer-Rao-regular families and in related linear or nonlinear regression cases. The conditions fail (and the estimator is inconsistent) in most discrete families.