# Some investigations in minimax estimation theory

• Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1985

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## Abstract

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1. IntroductionThough 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.CONTENTS1. Introduction.............................................................................................................................................................................................52. A theorem for a multinomial population...................................................................................................................................................53. Estimation of frequencies of population with a hierarchic structure........................................................................................................74. Estimation of frequencies of multinomial population under a general quadratic loss Function..............................................................125. A problem of prediction.........................................................................................................................................................................156. Minimax estimation of distribution function............................................................................................................................................177. Sequential minimax estimation for stochastic processes in the case where there exists a sufficient statistic for the parameter............188. Sequential estimation for the multinomial process.................................................................................................................................219. Exponential family of processes............................................................................................................................................................2310. Sequential estimation for a multivariate process.................................................................................................................................2411. Sequential estimation for the Poisson process....................................................................................................................................2912. Sequential minimax estimation in the case where the set of a priori distributions of the parameter is restricted.................................3113. Continuation to Section 12..................................................................................................................................................................3714. Final remarks......................................................................................................................................................................................40References...............................................................................................................................................................................................41

## How to cite

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Stanisław Trybuła. Some investigations in minimax estimation theory. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1985. <http://eudml.org/doc/268654>.

@book{StanisławTrybuła1985,
abstract = {1. IntroductionThough 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.CONTENTS1. Introduction.............................................................................................................................................................................................52. A theorem for a multinomial population...................................................................................................................................................53. Estimation of frequencies of population with a hierarchic structure........................................................................................................74. Estimation of frequencies of multinomial population under a general quadratic loss Function..............................................................125. A problem of prediction.........................................................................................................................................................................156. Minimax estimation of distribution function............................................................................................................................................177. Sequential minimax estimation for stochastic processes in the case where there exists a sufficient statistic for the parameter............188. Sequential estimation for the multinomial process.................................................................................................................................219. Exponential family of processes............................................................................................................................................................2310. Sequential estimation for a multivariate process.................................................................................................................................2411. Sequential estimation for the Poisson process....................................................................................................................................2912. Sequential minimax estimation in the case where the set of a priori distributions of the parameter is restricted.................................3113. Continuation to Section 12..................................................................................................................................................................3714. Final remarks......................................................................................................................................................................................40References...............................................................................................................................................................................................41},
author = {Stanisław Trybuła},
keywords = {multinomial populations; exponential family; sufficient statistics; quadratic loss function; prediction; minimax stopping rules; non- sequential models; minimax estimation},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Some investigations in minimax estimation theory},
url = {http://eudml.org/doc/268654},
year = {1985},
}

TY - BOOK
AU - Stanisław Trybuła
TI - Some investigations in minimax estimation theory
PY - 1985
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - 1. IntroductionThough 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.CONTENTS1. Introduction.............................................................................................................................................................................................52. A theorem for a multinomial population...................................................................................................................................................53. Estimation of frequencies of population with a hierarchic structure........................................................................................................74. Estimation of frequencies of multinomial population under a general quadratic loss Function..............................................................125. A problem of prediction.........................................................................................................................................................................156. Minimax estimation of distribution function............................................................................................................................................177. Sequential minimax estimation for stochastic processes in the case where there exists a sufficient statistic for the parameter............188. Sequential estimation for the multinomial process.................................................................................................................................219. Exponential family of processes............................................................................................................................................................2310. Sequential estimation for a multivariate process.................................................................................................................................2411. Sequential estimation for the Poisson process....................................................................................................................................2912. Sequential minimax estimation in the case where the set of a priori distributions of the parameter is restricted.................................3113. Continuation to Section 12..................................................................................................................................................................3714. Final remarks......................................................................................................................................................................................40References...............................................................................................................................................................................................41
LA - eng
KW - multinomial populations; exponential family; sufficient statistics; quadratic loss function; prediction; minimax stopping rules; non- sequential models; minimax estimation
UR - http://eudml.org/doc/268654
ER -

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