, Volume 42, Issue 1, pp 291–324 | Cite as

Necessary and sufficient conditions for consistency of generalizedM-estimates

  • Friedrich Liese
  • Igor Vajda


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.

Key words and phrases

Minimum contrast estimator M-estimator linear regression stationary ergodic observations consistency inconsistency 


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Copyright information

© Physica-Verlag 1995

Authors and Affiliations

  • Friedrich Liese
    • 1
  • Igor Vajda
    • 2
  1. 1.Department of MathematicsUniversity of RostockRostockGermany
  2. 2.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPragueCzech Republic

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