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Infinitezymalna odporność w bayesowskich modelach statystycznych
Języki publikacji
Abstrakty
Praca przedstawia niektóre z problemów dotyczących odporności (badania oscylacji wielkości a posteriori) procedur bayesowskich w wybranych klasach rozkładów a priori. Główna uwaga poświęcona jest odporności infinitezymalnej (ciągłość i różniczkowalność wielkości a posteriori), gdy w przestrzeni rozkladów a priori wprowadzona jest metryka Kolmogorowa. Omówiony jest też istotny problem wyznaczania procedur optymalnych.
The problem of measuring the Bayesian robustness is considered. An upper bound for the oscillation of a posterior functional in terms of the Kolmogorov distance between the prior distributions is given. The norm of the Frechet derivative as a measure of local sensitivity is presented. The problem of finding optimal statistical procedures is presented.
Wydawca
Rocznik
Tom
Strony
67--106
Opis fizyczny
Bibliogr. 50 poz.
Twórcy
autor
- Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa
Bibliografia
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- [3] Berger, J. O. (1980): Statistical decision theory. Fundations, Concepts, and Methods. Springer-Verlag.
- [4] Berger, J. O. (1984): The robust Bayesian viewpoint (with discussion). Robustness of Bayesian Analysis. J. Kadane, Ed. Amsterdam: North Holland, 63-124.
- [5] Berger, J. O. (1990): Robust Bayesian analysis: sensitivity to the prior. J. Statist. Plann. Inference 25, 303-328.
- [6] Berger, J. O i Berliner, L. M. (1986): Robust Bayes and empirical Bayes analysis with mi ε-contaminated priors. Ann. Statist. 14, 461-486.
- [7] Betro, B., Męczarski, M. i Ruggeri, F. (1992): Robust Bayesian analysis under generalized moments conditions. To appear in J. Statist. Plann. Inference.
- [8] Betro, B. i Ruggeri, F. (1992): Conditional T-minimax actions under convex losses. Commun. Statist. - Theory Meth. 21(4), 1051-1066.
- [9] Bickel, P. J. (1976): Another look at robustness: a review of Reviews and some new developements. Scand. J. Statist. 3, 145-168.
- [10] Boratyńska, A. (1992): Bayesian robustness in the e-contaninated class of priors. Komunikat na konferencji „International Workshop on Bayesian Robustness”, Milano, May 1992, złożone do druku.
- [11] Boratyńska, A. i Męczarski, M. (1994): Robust Bayesian estimation in the one-dimensional normal model, to appear in Statistics & Decisions 1994.
- [12] Boratyńska, A. i Zieliński, R. (1993): Bayesian robustness via Kolmogorov metric. Applicationes Math. 22, 139-143.
- [13] Bose, S. (1992): Bayesian robustness with local and nonlocal contaminations. An invited paper at „International Workshop on Bayesian Robustness”, Milano, May 1992.
- [14] Box, G. E. P. (1953): Non-normality and tests on variances. Biometrika 40, 318-335.
- [15] Box, G. E. P. i Andersen, S. L. (1955): Permutation theory in the derivation of robust criteria and the study of departures from assumptions. J. Roy. Statist. Soc. Ser. B 17, 1-34.
- [16] Clarke, B. R. (1983): Uniqueness and Frechet differentiability of functional solutions to maximum likelihood type equations. Ann. Statist. 11, 1196-1205.
- [17] Cuevas, A. i Sanz, P. (1988): On differentiability properties of Bayes operators. Bayesian Statistics 3. J. M. Bernardo, M. H. DeGroot, D. V. Lindley i A. F. M. Smith (eds), Oxford University Press, 569-577.
- [18] DasGupta, A. i Studden, W. J. (1988): Robust Bayesian analysis and optimal experimental designs in normal linear models with many parameters. Technical Report 88-14, Dept. of Stat. Purdue University.
- [19] DasGupta, A. i Studden, W. J. (1989): Frequentist behavior of robust Bayes estimates of normal means. Statistics & Decisions 7, 333-361.
- [20] DeRobertis, L. i Hartigan, J. A. (1981): Bayesian inference using intervals of measures. Ann. Statist. 9, 235-244.
- [21] Diaconis, P. i Freedman, D. (1986): On the consistency of Bayes estimates. Ann. Statist. 14, 1-67.
- [22] Fortini, S. i Ruggeri, F. (1990): Concentration function in a robust Bayesian frame work. Quaderno IAMI 90.6, Milano: CNR-IAMI.
- [23] Gelfand, A. E. i Dey, D. K. (1991): On Bayesian robustness of contaminated class of priors. Statistics & Decisions 9, 63-80.
- [24] Hampel, F. R., Ronchetti, E. M., Rousseeuw, R.J., Stahel, W. A. (1986): Robust statistics; the approach based on influence functions. Wiley, New York.
- [25] Huber, P. J. (1973): The use of Choquet capacities in statistics. Bull. Inst. Internat. Statist. 45, 181-191.
- [26] Huber, P. J. (1981): Robust statistics. Wiley, New York.
- [27] Kadane, J. B. i Chuang, D. T. (1978): Stable decision problems. Ann. Statist. 6, 1095-1110.
- [28] Lavine, M. (1991a): An approach to robust Bayesian analysis for multidimensional parameter spaces. JASA 86, 400-403.
- [29] Lavine, M. (1991b): Sensitivity in Bayesian statistics: the prior and the likelihood. JASA 86, 396-399.
- [30] Lavine, M., Wasserman, L., Wolpert, R. L. (1991): Bayesian inference with specified priors marginals. JASA 86, 964-971.
- [31] Learner, E. E. (1982): Sets of posterior means with bounded variance priors. Econometrica 50, 725-736.
- [32] Męczarski, M. (1993): Stable Bayesian estimation in the Poisson model. Zeszyty Nauk. PL - Sci. Bull. Łódź Technical Univ. No. 687-Matematyka, z. 25, 83-91.
- [33] Męczarski, M. (1993): Stability and conditional T-minimaxity in Bayesian inference. Applicationes Math. 22, 117-122.
- [34] Męczarski, M. i Zieliński, R. (1991): Stability of the Bayesian estimator of the Poisson mean under the inexactly specified gamma priors. Statistics & Probability Letters 12, 329-333.
- [35] Moreno, E. i Cano, J.A. (1991): Robust Bayesian analysis with ε-contaminations partially known. J. R. Statist. Soc. B 53, 143-155.
- [36] Moreno, E. i Pericchi, L. R. (1993): On ε-contaminated priors with quantile and picewise unimodality constraints. Commun. Statist. Theory Meth. 22(7), 1963-1978.
- [37] O’Hagan, A. i Berger, J. O. (1988): Ranges of posterior probabilities for quasi - unimodal priors with specified quantiles. JASA 83, 503-508.
- [38] Poetzelberger, K. i Polasek, W. (1991): Robust HPD-regions in Bayesian regression models. Econometrica 59, 1581-1589.
- [39] Rachev, S. T. (1991): Probability metrics and the stability of stochastic models. Wiley, Chichester.
- [40] Ruggeri, F. (1990): Posterior ranges of functions of parameters under priors with specified quantiles. Commun. Statist.—Theory Meth. 19, 127-144.
- [41] Ruggeri, F. (1991): Robust Bayesian analysis given a lower bound on the probability of a set. Commun. Statist.—Theory Meth. 20, 1881-1891.
- [42] Ruggeri, F. i Wasserman, L. (1991): Density based classes of priors: infinitesimal properties and approximations. Quaderno IAMI 91.4, CNR-IAMI, Milano.
- [43] Ruggeri, F. i Wasserman, L. (1993): Infinitesimal sensitivity of posterior distributions. Canadian J. Statist. 21(2), 195-203.
- [44] Salinetti, G. (1992): Stability of Bayesian decisions. An invited paper at „International Workshop on Bayesian Robustness”, Milano, May 1992.
- [45] Serfling, R. J. (1991): Twierdzenia graniczne statystyki matematycznej. PWN, Warszawa.
- [46] Sivaganesan, S. (1988): Range of posterior measures for priors with arbitrary contaminations. Commun. Statist.—Theory Meth. 17(5), 1591-1612.
- [47] Sivaganesan, S. and Berger, J. O. (1989): Ranges of posterior measures for priors with unimodal contaminations. Ann. Statist. 17, 868-889.
- [48] Wasserman, L. i Kadane, J. (1992): Computing bounds on expectations. JASA 87, 516-522.
- [49] Zieliński, R. (1983): Robust statistical procedures: a general approach. Lecture Notes in Mathematics 982, Springer-Verlag, 283-295.
- [50] Zolotarev, V.M. (1986): Contemporary Theory of Summation of Independent Random Variables. Nauka, Moscow. (In Russian).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-53ce4641-f7ee-40b7-859b-5457b23c3074