Means of a Dirichlet process and multiple hypergeometric functions

By Antonio Lijoi, E. Regazzini. 2004

64 citations recorded by the source corpus.

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov-Krein identity. Moreover, an expression of the characteristic function of the mean of a Dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth Lauricella function. This expression is then employed to prove that the distribution of the mean of a Dirichlet process is symmetric if and only if the parameter of the process is symmetric, and to provide a new expression of the moment generating function of the variance of a Dirichlet process.

Subfields: Applied Mathematics, Artificial Intelligence, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7dc4cc5e6a9aa99c596fe4fda493993c65c03b42

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Citations in the corpus

Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples (2007, background): “The study of the Dirichlet means can be traced back to an early work of Cifarelli and Regazzini [10] which culminates into the more recognized [11].”
Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “In Lijoi and Regazzini (2004) it is noted that when the baseline measure α is concentrated on a finite number of points, then the left–hand side of (47) coincides with the fourth Lauricella hypergeometric function.”
Asymptotic power of sphericity tests for high-dimensional data (2013, methodology): “The contour integral representation given in Lemma 2 has been derived independently by Mo (2012) and Wang (2012), who use it to study the largest sample covariance eigenvalue when the corresponding population eigenvalue equals the critical threshold or lies above it.”
Normalized random measures driven by increasing additive processes (2004, background): “We mention, among others, Muliere and Tardella (1998), Guglielmi and Tweedie (2001), Regazzini, Guglielmi and Di Nunno (2002) and Lijoi and Regazzini (2004) Received August 2002; revised July 2003.”
Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior (2005, background): “Other examples of this close connection can be found in Regazzini (1998) and Lijoi and Regazzini (2004), where functionals of the Dirichlet process are considered.”