scholarly journals A Compound Poisson Perspective of Ewens–Pitman Sampling Model

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2820
Author(s):  
Emanuele Dolera ◽  
Stefano Favaro
Keyword(s):  
Asymptotic Behaviour ◽  
Negative Binomial ◽  
Random Partitions ◽  
Compound Poisson ◽  
Poisson Sampling ◽  
Large N ◽  
Α Diversity ◽  
Analogous Compound ◽  
Special Case

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ>−α, or α<0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α<0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

MODELLING ZERO-INFLATED COUNT DATA WITH A SPECIAL CASE OF THE GENERALISED POISSON DISTRIBUTION

2019 ◽  
Vol 49 (03) ◽  
pp. 689-707 ◽  
Author(s):  
Enrique Calderín-Ojeda ◽  
Emilio GóMez-Déniz ◽  
Inmaculada Barranco-Chamorro
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Negative Binomial ◽  
Skewed Data ◽  
Iteratively Reweighted Least Squares ◽  
Number Of Zeros ◽  
Scoring Algorithm ◽  
Fisher Scoring Algorithm ◽  
Special Case ◽  
Excessive Number

AbstractA one-parameter version of the generalised Poisson distribution provided by Consul and Jain (1973) is considered in this paper. The distribution is unimodal with a zero vertex and over-dispersed. A generalised linear model related to this distribution is also presented. Its parameters can be estimated by using a Fisher-Scoring algorithm which is equivalent to iteratively reweighted least squares. Due to its flexibility and capacity to describe highly skewed data with an excessive number of zeros, the model is suitable to be applied in insurance settings as an alternative to the negative binomial and zero-inflated model.

On positive entire solutions of the elliptic equation Δu + K(x)up = 0 and its applications to Riemannian geometry

1996 ◽  
Vol 126 (2) ◽  
pp. 225-237 ◽  
Author(s):  
Changfeng Gui
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Scalar Curvature ◽  
Riemannian Geometry ◽  
Semilinear Elliptic Equation ◽  
Entire Space ◽  
Entire Solutions ◽  
Curvature Equation ◽  
Semilinear Elliptic ◽  
Special Case

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.

scholarly journals On the Distribution of the Surplus Prior and at Ruin

1999 ◽  
Vol 29 (2) ◽  
pp. 227-244 ◽  
Author(s):  
Hanspeter Schmidli
Keyword(s):  
Asymptotic Behaviour ◽  
Ruin Probability ◽  
Poisson Model ◽  
Subexponential Distributions ◽  
Compound Poisson ◽  
Initial Capital ◽  
Compound Poisson Model ◽  
Explicit Expressions

AbstractConsider a classical compound Poisson model. The safety loading can be positive, negative or zero. Explicit expressions for the distributions of the surplus prior and at ruin are given in terms of the ruin probability. Moreover, the asymptotic behaviour of these distributions as the initial capital tends to infinity are obtained. In particular, for positive safety loading the Cramer case, the case of subexponential distributions and some intermediate cases are discussed.

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

scholarly journals Asymptotic Behaviour of Compound Distributions

1985 ◽  
Vol 15 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Paul Embrechts ◽  
Makoto Maejima ◽  
Jozef L. Teugels
Keyword(s):  
Asymptotic Behaviour ◽  
Negative Binomial ◽  
Compound Distributions ◽  
Binomial Distributions

AbstractWe improve on some results of Sundt (1982) on the asymptotic behaviour of compound negative binomial distributions.

scholarly journals Optimizing the number of gates in quantum search

2017 ◽  
Vol 17 (3&4) ◽  
pp. 251-261
Author(s):  
Srinivasan Arunachalam ◽  
Ronald de Wolf
Keyword(s):  
Unique Solution ◽  
Search Algorithm ◽  
Quantum Search ◽  
Constant Number ◽  
Quantum Search Algorithm ◽  
Usual Form ◽  
Large N ◽  
Number Π ◽  
Special Case

In its usual form, Grover’s quantum search algorithm uses O( √ N) queries and O( √ N log N) other elementary gates to find a solution in an N-bit database. Grover in 2002 showed how to reduce the number of other gates to O( √ N log log N) for the special case where the database has a unique solution, without significantly increasing the number of queries. We show how to reduce this further to O( √ N log(r) N) gates for every constant r, and sufficiently large N. This means that, on average, the circuits between two queries barely touch more than a constant number of the log N qubits on which the algorithm acts. For a very large N that is a power of 2, we can choose r such that the algorithm uses essentially the minimal number π 4 √ N of queries, and only O( √ N log(log? N)) other gates.

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