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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

   
 
 

 

Distributional hyperspace-convergence of Argmin-sets in convex $M$-estimation


Author: Dietmar Ferger
Journal: Theor. Probability and Math. Statist. 109 (2023), 3-35
MSC (2020): Primary 60F05, 62E10; Secondary 60B05, 60B10
DOI: https://doi.org/10.1090/tpms/1195
Published electronically: October 3, 2023
MathSciNet review: 4652992
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Abstract: In $M$-estimation we consider the sets of all minimizing points of convex empirical criterion functions. These sets are random closed sets. We derive distributional convergence in the hyperspace of all closed subsets of the real line endowed with the Fell-topology. As a special case single minimizing points converge in distribution in the classical sense. In contrast to the literature so far, unusual rates of convergence and non-normal limits emerge, which go far beyond the square-root asymptotic normality. Moreover, our theory can be applied to the sets of zero-estimators.


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Additional Information

Dietmar Ferger
Affiliation: Fakultät Mathematik, Technische Universität Dresden, D-01069 Dresden, Germany
Email: dietmar.ferger@tu-dresden.de

Keywords: M-estimation, convex empirical processes, Argmin-sets, random closed sets, Fell-topology
Received by editor(s): April 12, 2022
Accepted for publication: November 21, 2022
Published electronically: October 3, 2023
Article copyright: © Copyright 2023 Taras Shevchenko National University of Kyiv