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## Some investigations in minimax estimation theory

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 240 wydano: 1985
Zawartość
Warianty tytułu
Abstrakty
EN

1. Introduction
Though the theory of minimax estimation was originated about thirty five years ago (see [7], [8], [9], [23]), there are still many unsolved problems in this area. Several papers have been devoted to statistical games in which the set of a priori distributions of the parameter was suitably restricted ([2], [10], [13]). Recently, special attention was paid to the problem of admissibility ([24], [3], [11], [12]).
This paper is devoted to the problem of determining minimax stopping rules and estimators in various situations. In Sections 2-6 nonsequential models and in Sections 7-13 sequential models are considered. In particular, sequential minimax estimation problems for the exponential class of processes are treated in detail.
The author intends to present a variety of situations rather than to consider them in their full generality, and thus some of the problems considered here admit further generalization.
EN

CONTENTS
1. Introduction.............................................................................................................................................................................................5
2. A theorem for a multinomial population...................................................................................................................................................5
3. Estimation of frequencies of population with a hierarchic structure........................................................................................................7
4. Estimation of frequencies of multinomial population under a general quadratic loss Function..............................................................12
5. A problem of prediction.........................................................................................................................................................................15
6. Minimax estimation of distribution function............................................................................................................................................17
7. Sequential minimax estimation for stochastic processes in the case where there exists a sufficient statistic for the parameter............18
8. Sequential estimation for the multinomial process.................................................................................................................................21
9. Exponential family of processes............................................................................................................................................................23
10. Sequential estimation for a multivariate process.................................................................................................................................24
11. Sequential estimation for the Poisson process....................................................................................................................................29
12. Sequential minimax estimation in the case where the set of a priori distributions of the parameter is restricted.................................31
13. Continuation to Section 12..................................................................................................................................................................37
14. Final remarks......................................................................................................................................................................................40
References...............................................................................................................................................................................................41
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 240
Liczba stron
42
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCXL
Daty
wydano
1985
Twórcy
autor
• Politechnika Wrocławska, Instytut Matematyki, Wrocław, Polska
• Institute of Mathematics, Technical University, Wrocław, Poland
Bibliografia
• [1] D. Blackwell, M. A. Girshick. Theory of games and statistical decisions. New York and London 1950.
• [2] J. R. Blum. J. Rosenblatt, Fixed precision estimation in the class of IFR distributions, Ann. Inst. Stat. Math. 21, I (1969).
• [3] L. D. Brown. A heuristic method for determining admissibility of estimator - with applications, Ann. Stat. 7 (1979), 960-994.
• [4] A. Dvoretzky, J. Kiefer, J. Wolfowitz, Sequential decision problems for processes with continuous time parameter. Problem of estimation, Ann. Math. Stat. 24 (1953), 403-415.
• [5] J. Franz, W. Winkler, Über Stoppzeiten bei statistischen Problemen für homogene Prozesse mit unabhängigen Zuwachsen, Math. Nachricht. 70 (1976), 37-53.
• [6] J. Franz, W. Winkler, Sequential estimation problems for the exponential class of processes with independent increments. Scand. Satist. 6 (1979), 129-139.
• [7] M. A. Girshick, L. J. Savage, Boyes and minimax estimates for quadratic loss functions. Proc. Second Berkeley Symp. Math. Statist. Prob. 1 (1951), 53-74.
• [8] J. L. Hodges, E. L. Lehmann, Some problems in minimax point estimation, Ann. Math. Stat. 21 (1950). 182-196.
• [9] J. L. Hodges, E. L. Lehmann, Some applications of the Cramer-Rao inequality, Proc. Second Berkeley Symp. Math. Statist. Prob. I (1951), 13-22.
• [10] J. L. Hodges, E. L. Lehmann, The use of previous experience in reaching statistical decisions, Ann. Math. Stat. 23 (1952), 376-407.
• [11] B. Johnson McK, On admissible estimators for certain fixed sample binomial problems, Ann. Math. Stat. 42 (1971). 1579 1587.
• [12] A. Kozek, Towards a calculus for admissibility, Ann. Stat. 10 (1982), 825-837.
• [13] H. Kudo, On partial prior information and the property of parametric sufficiency, Proc. Fifth Berkeley Symp. Math. Statist. Prob. 1 (1967), 251-265.
• [14] R. Magiera, On sequential minimax estimation for the exponential class of processes, Applicationes Mathematicae (Zastosowania Matematyki) 15 (1977), 445-454.
• [15] J. Olkin, M. Sobel, Admissible and minimax estimation for the multinomial distribution and for k independent binomial distributions, Ann. Stat. 7 (1979), 284-290.
• [16] E. S. Phadia, Minimax estimation of a cumulative distribution function, Ann. Stat. 1 (1973), 1149-1157.
• [17] R. Różański, A modification of Sudakov's lemma and efficient sequential plans for the Ornstein Uhlenbeck process, Applicationes Mathematicae (Zastosowania Matematyki) 17, 1 (1980), 73-85.
• [18] R. Różański,, On minimax sequential estimation of the mean value of a stationary Gaussian-Markov process, Applicationes Mathematicae (Zastosowania Matematyki) 17, 3 (1981), 25-32.
• [19] W. N. Sudakov, O merakh opredelyayemykh markovskimi momentami. Zapiski nauchnykh seminarov LOMI im. Steklova 12, Issledovaniya po teorii sluchaynykh processov (1969), 157 164 (in Russian).
• [20] S. Trybuła, Some problems in simultaneous minimax estimation, Ann. Math. Stat. 29 (1958), 245-53.
• [21] S. Trybuła, On some loss function, Colloq. Math. 7 (1960), 297-305.
• [22] S. Trybuła, Sequential estimation In processes with independent increments, Dissert. Math. 60 (1968), 1-49.
• [23] A. Wald, Statistical decision functions, New York 1950.
• [24] C. Stein, A necessary and sufficient condition for admissibility, Ann. Math. Statist. 26 (1955), 751-768.
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