5,518 research outputs found
On a Gibbs characterization of normalized generalized Gamma processes
We show that a Gibbs characterization of normalized generalized Gamma
processes, recently obtained in Lijoi, Pr\"unster and Walker (2007), can
alternatively be derived by exploiting a characterization of exponentially
tilted Poisson-Kingman models stated in Pitman (2003). We also provide a
completion of this result investigating the existence of normalized random
measures inducing exchangeable Gibbs partitions of type .Comment: 13 page
Approximating predictive probabilities of Gibbs-type priors
Gibbs-type random probability measures, or Gibbs-type priors, are arguably
the most "natural" generalization of the celebrated Dirichlet prior. Among them
the two parameter Poisson-Dirichlet prior certainly stands out for the
mathematical tractability and interpretability of its predictive probabilities,
which made it the natural candidate in several applications. Given a sample of
size , in this paper we show that the predictive probabilities of any
Gibbs-type prior admit a large approximation, with an error term vanishing
as , which maintains the same desirable features as the predictive
probabilities of the two parameter Poisson-Dirichlet prior.Comment: 22 pages, 6 figures. Added posterior simulation study, corrected
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Nested Archimedean Copulas Meet R: The nacopula Package
The package nacopula provides procedures for constructing nested Archimedean copulas in any dimensions and with any kind of nesting structure, generating vectors of random variates from the constructed objects, computing function values and probabilities of falling into hypercubes, as well as evaluation of characteristics such as Kendall's tau and the tail-dependence coefficients. As by-products, algorithms for various distributions, including exponentially tilted stable and Sibuya distributions, are implemented. Detailed examples are given.
Asymptotics for a Bayesian nonparametric estimator of species variety
In 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.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ371 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Rediscovery of Good-Turing estimators via Bayesian nonparametrics
The problem of estimating discovery probabilities originated in the context
of statistical ecology, and in recent years it has become popular due to its
frequent appearance in challenging applications arising in genetics,
bioinformatics, linguistics, designs of experiments, machine learning, etc. A
full range of statistical approaches, parametric and nonparametric as well as
frequentist and Bayesian, has been proposed for estimating discovery
probabilities. In this paper we investigate the relationships between the
celebrated Good-Turing approach, which is a frequentist nonparametric approach
developed in the 1940s, and a Bayesian nonparametric approach recently
introduced in the literature. Specifically, under the assumption of a two
parameter Poisson-Dirichlet prior, we show that Bayesian nonparametric
estimators of discovery probabilities are asymptotically equivalent, for a
large sample size, to suitably smoothed Good-Turing estimators. As a by-product
of this result, we introduce and investigate a methodology for deriving exact
and asymptotic credible intervals to be associated with the Bayesian
nonparametric estimators of discovery probabilities. The proposed methodology
is illustrated through a comprehensive simulation study and the analysis of
Expressed Sequence Tags data generated by sequencing a benchmark complementary
DNA library
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