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It is an elementary fact that the size of an orthogonal array of strength t on k factors must be a multiple of a certain number, say Lt, that depends on the orders of the factors. Thus Lt is a lower bound on the size of arrays of strength t on those factors, and is no larger than Lk, the size of the complete factorial design. We investigate the relationship between the numbers Lt, and two questions in particular: For what t is Lt < Lk? And when Lt = Lk, is the complete factorial design the only array of that size and strength t? Arrays are assumed to be mixed-level.We refer to an array of size less than Lk as a proper fraction. Guided by our main result, we construct a variety of mixed-level proper fractions of strength k ? 1 that also satisfy a certain group-theoretic condition. 相似文献
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Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics just to name a few)
and the bivariate Poisson distribution being a generalization of the Poisson distribution plays an important role in modelling
such data. In the present paper we present a Bayesian estimation approach for the parameters of the bivariate Poisson model
and provide the posterior distributions in closed forms. It is shown that the joint posterior distributions are finite mixtures
of conditionally independent gamma distributions for which their full form can be easily deduced by a recursively updating
scheme. Thus, the need of applying computationally demanding MCMC schemes for Bayesian inference in such models will be removed,
since direct sampling from the posterior will become available, even in cases where the posterior distribution of functions
of the parameters is not available in closed form. In addition, we define a class of prior distributions that possess an interesting
conjugacy property which extends the typical notion of conjugacy, in the sense that both prior and posteriors belong to the
same family of finite mixture models but with different number of components. Extension to certain other models including
multivariate models or models with other marginal distributions are discussed. 相似文献
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We develop a Bayesian framework for estimating the means of two random variables when only the sum of those random variables can be observed. Mixture models are proposed for establishing conjugacy between the joint prior distribution and the distribution for observations. Among other desirable features, conjugate distributions allow Bayesian methods to be applied in sequential decision problems. 相似文献
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