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
@book{bwmeta1.element.zamlynska-48fdf5f7-4764-4de6-9906-4c7a90c24b5c, author = {Stanis\l aw Trybu\l a}, title = {Some investigations in minimax estimation theory}, series = {GDML\_Books}, publisher = {Instytut Matematyczny Polskiej Akademi Nauk}, address = {Warszawa}, year = {1985}, zbl = {0572.62010}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-48fdf5f7-4764-4de6-9906-4c7a90c24b5c} }
Stanisław Trybuła. Some investigations in minimax estimation theory. GDML_Books (1985), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-48fdf5f7-4764-4de6-9906-4c7a90c24b5c/