Asymptotics for a Bayesian nonparametric estimator of species richness
AbstractIn Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two–parameter Poisson–Dirichlet process that was previously known only in the unconditional case.
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Bibliographic InfoPaper provided by University of Pavia, Department of Economics and Quantitative Methods in its series Quaderni di Dipartimento with number 144.
Length: 20 pages
Date of creation: May 2011
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Bayesian Nonparametrics; Species sampling models; Asymptotics; s–diversity; Polynomially and exponentially tilted random variables; Completely random measures; Normalized generalized gamma process; Two parameter Poisson–Dirichlet process.;
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- Antonio Lijoi & Igor Pruenster & Stephen G. Walker, 2008. "Bayesian nonparametric estimators derived from conditional Gibbs structures," ICER Working Papers - Applied Mathematics Series 06-2008, ICER - International Centre for Economic Research.
- Lancelot F. James & Antonio Lijoi & Igor Prünster, 2009. "Posterior Analysis for Normalized Random Measures with Independent Increments," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 36(1), pages 76-97.
- Argiento, Raffaele & Guglielmi, Alessandra & Pievatolo, Antonio, 2010. "Bayesian density estimation and model selection using nonparametric hierarchical mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 816-832, April.
- Stefano Favaro & Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2009. "Bayesian nonparametric inference for species variety with a two parameter Poisson-Dirichlet process prior," Carlo Alberto Notebooks 123, Collegio Carlo Alberto.
- Pierpaolo De Blasi & Giovanni Peccati & Igor Prünster, 2009. "Asymptotics for posterior hazards," Carlo Alberto Notebooks 122, Collegio Carlo Alberto.
- Lancelot F. James & Antonio Lijoi & Igor Prünster, 2006. "Conjugacy as a Distinctive Feature of the Dirichlet Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 33(1), pages 105-120.
- Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2007. "Bayesian Nonparametric Estimation of the Probability of Discovering New Species," Biometrika, Biometrika Trust, vol. 94(4), pages 769-786.
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