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.
References
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References
- N. Albrecht, Least squares estimation for binary decision trees, PhD thesis, Technische Universität Dresden, 2020.
- P. K. Andersen and R. D. Gill, Cox’s regression model for counting processes: a large sample study, Ann. Statist. 10 (1982), no. 4, 1100–1120. MR 0673646
- P. J. Bickel, C. A. Klaassen, Y. Ritov and J. A. Wellner, Efficient and adaptive inference in semiparametric models, Springer, New York, 1998. MR 1623559
- P. Billingsley, Convergence of probability measures, John Wiley & Sons, New York, 1968. MR 0233396
- D. Ferger, A continuous mapping theorem for the Argmin-set functional with applications to convex stochastic processes, Kybernetika 57 (2021), no. 3, 426–445. MR 4299457
- B. Fristedt and L. Gray, A modern approach to probability theory, Probability and its Applications, Birkhäuser, Boston, 1997. MR 1422917
- P. Gänssler and W. Stute, Wahrscheinlichkeitstheorie, Springer-Verlag, Berlin-New York, 1977. MR 0501219
- S. J. Habermann, Concavity and estimation, Ann. Statist. 17 (1989), no. 4, 1631–1661. MR 1026303
- N. L. Hjort, Bayes estimators and asymptotic efficiency in parametric counting process models, Scand. J. Statist. 13 (1986), no. 2, 63–85. MR 0867355
- N.L. Hjort and D. Pollard, Asymptotic for minimizers of convex processes, Preprint, Dept. of Statistics, Yale University, 1993. http://arxiv.org/abs/1107.3806v1.
- P. Huber, Robust estimation of a location parameter, Ann. Math. Statist. 35 (1964), 73–101. MR 0161415
- P. Huber, Robust statistics, John Wiley & Sons, New York, 1981. MR 0606374
- J. Jurečková, Asymptotic relations of M-estimates and R-estimates in linear regression model, Ann. Statist. 5 (1977), no. 3, 464–472. MR 0433698
- J. Jurečková, Estimation in a linear model based on regression rank scores, J. Nonparametr. Statist. 1 (1992), no. 3, 197–203. MR 1241522
- K. Knight, What are the limiting distributions of quantile estimators? Statistical data analysis based on the $L_1$-norm and related methods (Neuchâtel, 2002) (Y. Dodge, ed.), Stat. Ind. Technol, Birkhäuser, Basel, 2002, pp. 47–65. MR 2001304
- F. Liese and K-J. Mieschke, Statistical decision theory, Springer Series in Statistics, Springer, New York, 2008. MR 2421720
- I. Molchanov, Theory of random sets, Probability Theory and Stochastic Modelling, vol. 87, Springer, London, 2017. MR 3751326
- H. T. Nguyen, An introduction to random sets, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2220761
- C. P. Niculescu and L.-E. Persson, Convex functions and their applications: A contemporary approach, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 23, Springer, New York, 2006. MR 2178902
- W. Niemiro, Asymptotic for M-estimators defined by convex minimization, Ann. Statist. 20 (1992), no. 3, 1514–1533. MR 1186263
- T. Norberg, Convergence and existence of random set distributions, Ann. Probab. 12 (1984), no. 3, 726–732. MR 0744229
- H. Nyquist, The optimal $L_p$ norm estimator in linear regression models, Comm. Statist. A—Theory Methods 12 (1983), no. 21, 2511–2524. MR 0715180
- J. Pfanzagl, Parametric statistical theory, De Gruyter Textbook, De Gruyter, Berlin, 1994. MR 1291393
- D. Pollard, Empirical processes: Theory and applications, NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 2, Inst. Math. Statist., Hayward, CA, 1990. MR 1089429
- D. Pollard, Asymptotics for least absolute deviation regression estimators, Econometric Theory 7 (1991), no. 2, 186–199. MR 1128411
- R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, vol. 28, Princeton Univ. Press, Princeton, NJ, 1970. MR 0274683
- R. T. Rockafellar and R. J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften, vol. 317, Springer, Berlin, 1998. MR 1491362
- R. J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1980. MR 0595165
- N. V. Smirnov, Limit distributions for the terms of a variational series, Amer. Math. Soc. Translation, vol. 67, 1952. MR 0047277
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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
