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Necessary and sufficient conditions for consistency of generalizedM-estimates
Authors:Friedrich Liese  Igor Vajda
Institution:(1) Department of Mathematics, University of Rostock, Universitätsplatz 1, 18055 Rostock, Germany;(2) Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 18208 Prague, Czech Republic
Abstract:GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximateM-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximateM-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximateM-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency ofM-estimators known from the literature. A linear regression example is presented where theM-estimate is consistent and an approximateM-estimate is incosistent.Supported by CSAS grant N. 17503.
Keywords:Minimum contrast estimator  M-estimator  linear regression  stationary ergodic observations  consistency  inconsistency
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