Abstract
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.
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REFERENCES
H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.
H. Attouch and R. Wets, Quantitative stability of variational systems: 1. The epigraphical distance, Trans. Am. Math. Soc. 328, 695-729 (1991).
S. Dolecki, G. Salinetti, and R. Wets, Convergence of functions: equisemicontinuity, Trans. Am. Math. Soc. 276, 409-429 (1983).
J. Dupacova and R. Wets, Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems, Ann. Stat. 16, 1517-1549 (1988).
P. J. Huber, The behavior of maximum likelihood estimates under nonstandard conditions, in Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 1967, pp. 221-233.
S. T. Rachev, Levy-Prohorov distance in a space of semicontinuous set functions, in Stability Problems for Stochastic Models, VNIISI, Moscow, 1980, pp. 76-88 (in Russian) (English translation: J. Sov. Math. 32, 64–74 (1986)).
S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, New York, 1991.
R. T. Rockafellar and R. Wets, Variational systems: an introduction, multifunctions and integrands, in Approximation and Optimization (G. Salinetti, ed.), Lecture notes in mathematics No. 1091, Springer-Verlag, Berlin, 1984, pp. 1-54.
G. Salinetti, On structural relationships between probability and statistics: empirical processes and statistical functionals, Atti della XXXV Riunione Scientifica della Società Italiana di Statistica, Vol. 1, 1990, pp. 75-96 (in Italian).
G. Salinetti and R. Wets, On the convergence of closed-valued multifunctions, Trans. Am. Math. Soc. 266, 275-289 (1981).
W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5, 67-85 (1980).
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Dall'Aglio, M., Rachev, S.T. Metrization of Epi-Convergence: An Application to the Strong Consistency of M-Estimators. Journal of Computational Analysis and Applications 1, 63–86 (1999). https://doi.org/10.1023/A:1022618620244
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DOI: https://doi.org/10.1023/A:1022618620244