EMPIRICAL BAYES RULES FOR SELECTING THE BEST NORMAL POPULATION COMPARED WITH A CONTROL
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In this paper, problems of sequential decision theory are taken into consideration by extending the definition of the BAYES rule and treating BAYES rules. This generalisation is quite useful for practice. In many cases only BAYES rules can be calculated. The conditions under which such sequential decision procedures exist are demonstrated, as well as how to construct them on a scheme of backward induction resulting in the conclusion that the existence of BAYES rules needs essentially weaker assumptions than the existence of BAYES rules.Futhermore, methods are searched to simplify the construction of optimal stopping rules. Some illustrative examples are given.
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We study the empirical Bayes approach to the sequential estimation problem. An empirical Bayes sequential decision procedure, which consists of a stopping rule and a terminal decision rule, is constructed for use in the component. Asymptotic behaviors of the empirical Bayes risk and the empirical Bayes stopping times are investigated as the number of components increase.
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Two-parameter models of fractional statistics aimed at finding an expression for the occupation numbers of free anyons have been considered. Virial coefficients are found for statistics of several types: к-deformed Polychronakos and Haldane–Wu statistics, Polychronakos and Haldane–Wu statistics modified with the q-exponential in the bosonic limit, and incomplete and nonadditive Gentile statistics for various level-filling maxima. A relation between the anyonic statistics and various statistics of fractional types is found and analyzed.
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We made in this paper a brief analysis of the following statistics: Intermediate Statistics, Parastatistics, Fractionary Statistics and Gentileonic Statistics that predict the existence of particles which are different from bosons, fermions and maxwellons. We shown the fundamental hypothesis assumed in each one of the above mentioned statistics and their main predictions and compared them with experimental results. Taking into account the works done about these statistics we could say that there is a tendency to believe that real particles, that is, those that can be observed freely, can be only bosons, fermions and maxwellons and that all other particles, different from these would be quasiparticles. Up to date in 3-dim systems only bosons, fermions and maxwellons have been detected freely. Recently in 2-dim systems have been detected the quasiparticles named anyons that have fractionary charges and spins.
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Abstract A Bayesian approach is given for various kinds of empirical Bayes problems. In particular it is shown that empirical Bayes procedures are really non-Bayesian, asymptotically optimal, classical procedures for mixtures. In some situations these procedures are Bayes with respect to some prior and in other situations, there is no prior for which they are Bayes. Several examples of these concepts are given as well as a general theory showing the difference between an empirical Bayes model and a Bayes empirical Bayes model.
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10.1002/9780470317105.ch4.abs The prelims comprise: Introduction General Form of Bayes' Theorem for Events Bayes' Theorem for Discrete Data and Discrete Parameter Bayes' Theorem for Continuous Data and Discrete Parameter Bayes' Theorem for Discrete Data and Continuous Parameter Bayes' Theorem for Discrete Data and Continuous Parameter
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Abstract Bayes’ theorem is 250 years old this year. But did the Rev. Thomas Bayes actually devise it? Martyn Hooper presents the case for the extraordinary Richard Price, friend of US presidents, mentor, pamphleteer, economist, and above all preacher. And did Price develop Bayes’ theorem in order to prove the existence of God?
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