×

zbMATH — the first resource for mathematics

Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator. (English) Zbl 0862.62029
Summary: Using the variational properties of epi-convergence together with suitable results on the measurability of multifunctions and integrands, we prove a strong law of large numbers for sequences of integrands form which we deduce a general theorem of almost sure convergence (strong consistency) for the maximum likelihood estimator.

MSC:
62F12 Asymptotic properties of parametric estimators
60F15 Strong limit theorems
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
49K45 Optimality conditions for problems involving randomness
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ARSENIN, W. J., LJAPUNOV, A. A. and CEGOLKOV, E. A. 1955. Arbeiten zur deskriptiven mengenlehre. Deutscher Verlag der Wissenschaften, Berlin. · Zbl 0068.27001
[2] ATTOUCH, H. 1984. Variational Convergence for Functions and Operators. Pitman, Boston. · Zbl 0561.49012
[3] ATTOUCH, H. and WETS, R. 1991. Epigraphical processes: laws or large numbers for random lsc functions. Expose No. 13, Seminaire d’Analy se Convexe, Univ. Montpellier, \' \' France. · Zbl 0744.60021
[4] BAHADUR, R. R. 1958. Examples of inconsistency of maximum likelihood estimates. Sankhy a Ser. A 20 207 210. · Zbl 0087.34202
[5] BEER, G. 1993. Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications. Kluwer, Dordrecht. · Zbl 0792.54008
[6] BROWN, L. D. and PURVES, R. 1973. Measurable selections of extrema. Ann. Statist. 1 902 912. · Zbl 0265.28003
[7] CASTAING, C. 1976. A propos de l’existence des sections separement mesurables et separe\' \' \' ḿent continues d’une multiapplication separement semi continue inferieurement. \' \' \' Travaux Sem. Anal. Convexe 6. \' · Zbl 0356.46045
[8] CASTAING, C. and VALADIER, M. 1977. Convex Analy sis and Measurable Multifunctions. Lecture Notes in Math. 580. Springer, New York. · Zbl 0346.46038
[9] DELLACHERIE, C. 1975. Ensembles analytiques: theoremes de separation et applications. \' Śeminaire de Probabilites IX. Lecture Notes in Math. 465. Springer, Berlin. \' · Zbl 0354.54023
[10] DUPACOVA, J. and WETS, R. 1988. Asy mptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. 16 1517 1549. · Zbl 0667.62018
[11] ETEMADI, N. 1981. An elementary proof of the strong law of large numbers. Z. Wahrsch. Verw. Gebiete 55 119 122. · Zbl 0438.60027
[12] HESS, C. 1991. Convergence of conditional expectations for unbounded random sets. Math. Oper. Res. 16 627 649. JSTOR: · Zbl 0744.28010
[13] HESS, C. 1991. On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence. J. Multivariate Anal. 39 175 201. · Zbl 0746.60051
[14] HESS, C. 1991. Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator. Cahier de Mathematique de la Decision, Univ. Paris Dauphine. 9121. JSTOR: · Zbl 0744.28010
[15] HIMMELBERG, C. J. 1975. Measurable relations. Fund. Math. 87 53 72. · Zbl 0296.28003
[16] HIMMELBERG, C. J., PARTHASARATHY, T. and VAN VLECK, F. S. 1981. On measurable relations. Fund. Math. 111 161 167. · Zbl 0465.28002
[17] IBRAGIMOV, I. A. and HAS’MINSKI, R. Z. 1981. Statistical Estimation, Asy mptotic Theory. Springer, New York.
[18] JALBY, V. 1993. Semi-continuite, convergence et approximation des applications vectoŕielles, Loi des grands nombres. Preprint. Laboratoire d’Analy se Convexe, Univ. Montpellier.
[19] KING, A. J. and WETS, R. 1991. Epi-consistency of convex stochastic programs. Stochastics Stochastics Rep. 34 83 92. · Zbl 0733.90049
[20] LANERY, E. 1991. Convergence presque sure du maximum de vraisemblance. C. R. Acad. Ści. Paris Ser. I Math. 313 301 303. \' · Zbl 0734.60032
[21] MOSCO, U. 1969. Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3 510 585. · Zbl 0192.49101
[22] PENOT, J. P. and POMMELET, A. 1985. Toward minimal assumptions for the infimal convolution regularization. Technical Report 23, AVAMAC, Univ. Perpignan.
[23] ROCKAFELLAR, R. T. 1976. Integral functionals, normal integrands and measurable selections. Non-linear Operators and Calculus of Variations. Lecture Notes in Math. 543. Springer, Berlin. · Zbl 0374.49001
[24] ROCKAFELLAR, R. T. and WETS, R. 1984. Variational sy stems, an introduction. In MultiZ functions and Integrands, Stochastic Analy sis, Approximation and Optimization G.. Salinetti, ed.. Lecture Notes in Math. 1091 1 54. Springer, Berlin.
[25] WALD, A. 1949. Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595 601. · Zbl 0034.22902
[26] WIJSMAN, R. A. 1966. Convergence of sequences of convex sets. II. Trans. Amer. Math. Soc. 123 32 45. · Zbl 0146.18204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.