TY - JOUR
T1 - Metrization of Epi-Convergence
T2 - An Application to the Strong Consistency of M-Estimators
AU - Dall'Aglio, Marco
AU - Rachev, Svetlozar T.
N1 - Funding Information:
This work was carried out while M. Dall'Aglio was visiting the Department of Statistics and Applied Probability, University of California, Santa Barbara during the spring quarter 1993 and the spring quarter 1996. The last visit was supported by C.N.R. grant no. 203.10.31. Grateful acknowledgment is made for hospitality.
PY - 1999
Y1 - 1999
N2 - 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.
AB - 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.
KW - Consistency
KW - Convergence of infima
KW - Epi-convergence
KW - Hausdorff distance
KW - M-estimators
UR - http://www.scopus.com/inward/record.url?scp=0346366117&partnerID=8YFLogxK
U2 - 10.1023/A:1022618620244
DO - 10.1023/A:1022618620244
M3 - Article
AN - SCOPUS:0346366117
SN - 1521-1398
VL - 1
SP - 63
EP - 86
JO - Journal of Computational Analysis and Applications
JF - Journal of Computational Analysis and Applications
IS - 1
ER -