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21
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Generalized weighted Chinese restaurant processes for species sampling mixture models
 Statistica Sinica
, 2003
"... Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conj ..."
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Cited by 53 (8 self)
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Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its twoparameter extension, the PitmanYor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, twoparameter PoissonDirichlet process, prediction rule, random probability measure, species sampling sequence.
Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived ..."
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Cited by 24 (1 self)
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
ASYMPTOTIC BEHAVIOR OF THE POISSON–DIRICHLET DISTRIBUTION FOR LARGE MUTATION RATE 1
, 2006
"... The large deviation principle is established for the Poisson–Dirichlet distribution when the parameter θ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson–Dirichlet distribution with selection. A phase transition occurs depending on the ..."
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Cited by 12 (5 self)
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The large deviation principle is established for the Poisson–Dirichlet distribution when the parameter θ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson–Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in θ, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in θ, then the large deviation rate function includes an additional term coming from selection. The application of these results to the heterozygote advantage model provides an alternate proof of one of Gillespie’s conjectures in [Theoret.
New strings for old Veneziano amplitudes III. Symplectic treatment
, 2005
"... A d−dimensional rational polytope P is a polytope whose vertices are located at the nodes of Z d lattice. Consider the number ∣ ∣kP ∩ Z d ∣ ∣ of points inside the inflated P with coefficient of inflation k (k = 1, 2, 3,...). The Ehrhart polynomial of P counts the number of such lattice points insid ..."
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Cited by 10 (3 self)
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A d−dimensional rational polytope P is a polytope whose vertices are located at the nodes of Z d lattice. Consider the number ∣ ∣kP ∩ Z d ∣ ∣ of points inside the inflated P with coefficient of inflation k (k = 1, 2, 3,...). The Ehrhart polynomial of P counts the number of such lattice points inside the inflated P and (may be) at its faces (including vertices). In Part I [ JGP 55 (2005) 50] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) physical models reproducing the Veneziano (and Venezianolike) amplitudes. General ideas (e.g. those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on ππ scattering, etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [PNAS
The PoissonDirichlet Distribution And Its Relatives Revisited
, 2001
"... The PoissonDirichlet distribution and its marginals are studied, in particular the largest component, that is Dickman's distribution. Sizebiased sampling and the GEM distribution are considered. Ewens sampling formula and random permutations, generated by the Chinese restaurant process, are also i ..."
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Cited by 10 (0 self)
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The PoissonDirichlet distribution and its marginals are studied, in particular the largest component, that is Dickman's distribution. Sizebiased sampling and the GEM distribution are considered. Ewens sampling formula and random permutations, generated by the Chinese restaurant process, are also investigated. The used methods are elementary and based on properties of the finitedimensional Dirichlet distribution. Keywords: Chinese restaurant process; Dickman's function; Ewens sampling formula; GEM distribution; Hoppe's urn; random permutations; residual allocation models; sizebiased sampling ams 1991 subject classification: primary 60g57 secondary 60c05, 60k99 Running title: The PoissonDirichlet distribution revisited 1
The PoissonDirichlet distribution and the scaleinvariant Poisson process
 COMBIN. PROBAB. COMPUT
, 1999
"... We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that ..."
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Cited by 8 (1 self)
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We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T � 1. Restricting both processes to (0,β] for 0 <β � 1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.
Some Superpopulation Models for Estimating the Number of Population Uniques
, 1997
"... The number of the unique individuals in the population is of great importance in evaluating the disclosure risk of a microdata set. We approach this problem by considering some basic superpopulation models including the gammaPoisson model of Bethlehem et al. (1990). We introduce Dirichletmultinomi ..."
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Cited by 7 (5 self)
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The number of the unique individuals in the population is of great importance in evaluating the disclosure risk of a microdata set. We approach this problem by considering some basic superpopulation models including the gammaPoisson model of Bethlehem et al. (1990). We introduce Dirichletmultinomial model which is closely related but more basic than the gammaPoisson model, in the sense that binomial distribution is more basic than Poisson distribution. We also discuss the Ewens model and show that it can be obtained from the Dirichletmultinomial model by a limiting argument similar to the law of small numbers. The multivariate Ewens distribution is a basic mathematical model used in genetics. Estimation of the number of the population uniques is particularly simple under the Ewens model. Although these models might not necessarily well fit actual populations, they can be considered as basic mathematical models for our problem, as binomial and Poisson distributions are considered as...
A twoparameter family infinitedimensional diffusions in the Kingman simplex
, 2007
"... The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex ..."
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Cited by 5 (1 self)
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The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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Cited by 3 (0 self)
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1