Abstract:The purpose of this paper is twofold: First, to show that a natural enlargement of the class of natural conjugate priors-namely, mixtures of natural conjugate priors-also leads to mathematically tractable solutions. Secondly, to show that this enlargement is "adequate" in that any prior may be arbitrarily closely approximated by a suitable member of this class.Specifically, a general method for approximating an arbitrary prior density is first given with approximation (convergence) defined pointwise or in the … Show more
“…Based on theoretical results by De Vore e Lorenz (1993) and Asmussen (1987), Wiper et al (2001) used mixtures of Gamma densities to approximate any density defined over [0, ∞). See also Dalal and Hall (1983) and Dey et al (1995) for related work.…”
Section: Nonparametric Estimation Of Curvesmentioning
This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.
“…Based on theoretical results by De Vore e Lorenz (1993) and Asmussen (1987), Wiper et al (2001) used mixtures of Gamma densities to approximate any density defined over [0, ∞). See also Dalal and Hall (1983) and Dey et al (1995) for related work.…”
Section: Nonparametric Estimation Of Curvesmentioning
This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.
“….}. Her work extended from the results of Dalal and Hall (1983) and Diaconis and Ylvisaker (1985) who proved that for sufficiently large p, mixtures of the form given in (2) can approximate any c.d.f. on [0, 1] to any arbitrary degree of accuracy.…”
Section: Random Bernstein Polynomial Priormentioning
“…Appropriate choice of the mixing probabilities and the gamma density parameters can provide arbitrarily close approximation (Dalal and Hall 1983) to a wide range of possible scenarios regarding the dependence among θ 's. In general the literature regarding multivariate mixtures of gamma densities is sparse (we are aware of the work of Gaver 1970 in the general setting).…”
Section: The Prior Distributionmentioning
confidence: 98%
“…Dependent prior distributions can be quite flexible since they offer the opportunity to represent special patterns. We define and base our inference on a prior which is a finite mixture of conditionally independent gamma densities and extend the idea of mixtures of conjugate priors (Dalal and Hall 1983). Both the joint and marginal posterior distributions can be obtained as a finite mixture of conditionally independent gamma distributions, i.e.…”
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.
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