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A large number of statistical models is described by a family of reversed submartingales converging to degenerated limits. The problem under consideration is to estimate the maximum points of the limit function. For this, various maximum functions are used and consequently different concepts of consistency are introduced. In this paper we introduce and investigate a general reversed submartingale framework for these models. Our approach relies upon the i.i.d. case [6]. We show that the best known sufficient conditions for consistency in this case remain valid for conditionally S-regular families of reversed submartingales introduced in [13], which are known to include all 17-processes. Moreover, by using results on uniform convergence of families of reversed submartingales [15], we deduce new conditions for consistency. These conditions are expressed by means of Hardy’s regular convergence [4], and are of a total boundedness in the mean type. In this way the problem of consistency is naturally connected with the infinitely dimensional (uniform) reversed submartingale convergence theorem. Applications to a stochastic maximization of families of random processes over time sets are also given.
Czasopismo
Rocznik
Tom
Strony
289--318
Opis fizyczny
Bibliogr. 18 poz.
Twórcy
autor
- Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark
- Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
Bibliografia
- [1] H. Cramér, Mathematical Methods of Statistics, Princeton University Press, 1946.
- [2] R. A. Fisher, On the mathematical foundations of theoretical statistics, Philos. Trans. Roy. Soc. London, Ser. A, Vol. 22 (1922), pp. 309-368.
- [3] — Theory of statistical estimation, Proc. Cambridge Philos. Soc. 22 (1925), pp. 700-725.
- [4] G. H. Hardy, On the convergence of certain multiple series, Math. Proc. Cambridge Philos. Soc. 19 (1917), pp. 86-95.
- [5] J. Hoffmann-Jørgensen, Pointwise compact metrizable sets of functions and consistency of statistical models, Institute of Mathematics, University of Aarhus, Preprint Series No. 15 (1991), 6 pp.
- [6] — Asymptotic likelihood theory, in: Functional Analysis III, Proc. Conf. Dubrovnik 1989, Various Publ. Series No. 40 (1992), pp. 7-192.
- [7] V. S. Huzurbazar, The likelihood equation, consistency and the maxima of the likelihood function, Annals of Eugenics 14 (1948), pp. 185-200.
- [8] J. Kiefer and J. Wolfowitz, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters, Ann. Math. Statist. 27 (1956), pp. 887-906.
- [9] L. Le Cam and L. Schwartz, A necessary and sufficient condition for the existence of consistent estimates, ibidem 31 (1960), pp. 140-150.
- [10] R. H. Norden, A survey of maximum likelihood equation, Internat. Statist. Rev. 40 (1972), pp. 329-354.
- [11] — A survey of maximum likelihood (Part 2), ibidem 41 (1972), pp. 39-58.
- [12] G. Peškir, Measure compact sets of functions and consistency of statistical models, Theory Probab. Appl. 38 (1993), pp. 360-367.
- [13] — On separability of families of reversed submartingales, Proc. Conf. Probability in Banach Spaces IX (Sandbjerg 1993), Birkhäuser, Boston 1994, pp. 36-53.
- [14] — The existence of measurable approximating máximums, Math. Scand. 77 (1995), pp. 71-84.
- [15] — Uniform convergence of reversed martingales, J. Theoret. Probab. 8 (1995), pp. 387-415.
- [16] E. J. G. Pitman, Some Basic Theory for Statistical Inference, Chapman and Hall, London 1979.
- [17] A. Wald, Note on the consistency of the maximum likelihood estimate, Ann. Math. Statist. 20 (1949), pp. 595-601.
- [18] J. Wolfowitz, On Wald’s proof of the consistency of the maximum likelihood estimate, ibidem 20 (1949), pp. 601-602.
Typ dokumentu
Bibliografia
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