Metrization of Epi-Convergence: An Application to the Strong Consistency of M-Estimators

Marco Dall'Aglio, Svetlozar T. Rachev

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Strong consistency in the class of M-estimators is examined here as an application of epi-convergence, a functional convergence which is particularly suited for the study of convergence of the functions' minimizing values and arguments. Starting from a 1988 paper by J. Dupačova and R. Wets, which contains a thorough account of the relations between consistency and epi-convergence, a quantitative approach of the same topic is pursued here. Epi-convergence is compared with two definitions introduced in 1980 by one of the authors. The results are merged in order to define a distance between lower semicontinuous functions that is compatible with epi-convergence and bounds the distance between the minimizing arguments. These results applied to the statistical problem allow the definition of a bound of the distance between the estimator and the parameter.

Original languageEnglish
Pages (from-to)63-86
Number of pages24
JournalJournal of Computational Analysis and Applications
Issue number1
StatePublished - 1999


  • Consistency
  • Convergence of infima
  • Epi-convergence
  • Hausdorff distance
  • M-estimators


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