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1
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Conjugate priors for exponential-type processes with random initial conditions

100%
Magiera R.
Applicationes Mathematicae
|
1993-1995
|
tom 22
|
nr 3
321-330
EN
The family of proper conjugate priors is characterized in a general exponential model for stochastic processes which may start from a random state and/or time.
2
Content available remote

Bayes sequential estimation procedures for exponential-type processes

100%
Magiera R.
Applicationes Mathematicae
|
1993-1995
|
tom 22
|
nr 3
311-320
EN
The Bayesian sequential estimation problem for an exponential family of processes is considered. Using a weighted square error loss and observing cost involving a linear function of the process, the Bayes sequential procedures are derived.
3
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Γ-minimax sequential estimation for Markov-additive processes

100%
Magiera R.
Applicationes Mathematicae
|
2001
|
tom 28
|
nr 4
467-485
EN
The problem of estimating unknown parameters of Markov-additive processes from data observed up to a random stopping time is considered. To the problem of estimation, the intermediate approach between the Bayes and the minimax principle is applied in which it is assumed that a vague prior information on the distribution of the unknown parameters is available. The loss in estimating is assumed to consist of the error of estimation (defined by a weighted squared loss function) as well as a cost of observing the process up to a stopping time. Several classes of optimal sequential procedures are obtained explicitly in the case when the available information on the prior distribution is restricted to a set Γ which is determined by certain moment-type conditions imposed on the prior distributions.
4
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On minimax sequential procedures for exponential families of stochastic processes

100%
Magiera R.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 1
1-18
EN
The problem of finding minimax sequential estimation procedures for stochastic processes is considered. It is assumed that in addition to the loss associated with the error of estimation a cost of observing the process is incurred. A class of minimax sequential procedures is derived explicitly for a one-parameter exponential family of stochastic processes. The minimax sequential procedures are presented in some special models, in particular, for estimating a parameter of exponential families of diffusions, for estimating the mean or drift coefficients of the class of Ornstein-Uhlenbeck processes, for estimating the drift of a geometric Brownian motion and for estimating a parameter of a family of counting processes. A class of minimax sequential rules for a compound Poisson process with multinomial jumps is also found.
5
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Wpływ Profesora Stanisława Trybuły na rozwój teorii estymacji sekwencyjnej dla procesów stochastycznych

64%
Magiera R., Wilczyński M.
Mathematica Applicanda
|
2010
|
tom 38
|
nr 1
PL
 W artykule omówiono wkład Stanisława Trybuły w badania dotyczące sekwencyjnej estymacji dla procesów stochastycznych. Dwa jego artykuły, opublikowane w Dissertationes Mathematicae (1968, 1985), miały istotny wpływ na przyszły rozwój tej dziedziny i zainteresowały wielu statystyków matematyków. W artykule pokrótce omawiamy rezultaty uzyskane przez autorów zainspirowanych tymi dwoma fundamentalnymipracami Stanisława Trybuły.Słowa kluczowe: estymacja sekwencyjna, efektywny plan sekwencyjny, proces stochastyczny, minimaksowy plan sekwencyjny, wykładnicza rodzina procesów
6
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The Bayes choice of an experiment in estimating a success probability

64%
Jokiel-Rokita A., Magiera R.
Applicationes Mathematicae
|
2002
|
tom 29
|
nr 2
135-144
EN
A Bayesian method of estimation of a success probability p is considered in the case when two experiments are available: individual Bernoulli (p) trials-the p-experiment-or products of r individual Bernoulli (p) trials-the $p^{r}$-experiment. This problem has its roots in reliability, where one can test either single components or a system of r identical components. One of the problems considered is to find the degree r̃ of the $p^{r̃}$-experiment and the size m̃ of the p-experiment such that the Bayes estimator based on m̃ observations of the p-experiment and N-m̃ observations of the $p^{r̃}$-experiment minimizes the Bayes risk among all the Bayes estimators based on m observations of the p-experiment and N-m observations of the $p^{r}$-experiment. Another problem is to sequentially select some combination of these two experiments, i.e., to decide, using the additional information resulting from the observation at each stage, which experiment should be carried out at the next stage to achieve a lower posterior expected loss.
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