A generalised Dickman distribution and the number of species in a negative binomial process model

2021 ◽  
Vol 53 (2) ◽  
pp. 370-399
Author(s):  
Yuguang Ipsen ◽  
Ross A. Maller ◽  
Soudabeh Shemehsavar
Keyword(s):  
Poisson Process ◽  
Process Model ◽  
Negative Binomial ◽  
Sample Distribution ◽  
Large Sample ◽  
Number Of Species ◽  
Dirichlet Model ◽  
Negative Binomial Process ◽  
Binomial Process ◽  
Two Parameter

AbstractWe derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals On the long-range dependence of fractional Poisson and negative binomial processes

2016 ◽  
Vol 53 (4) ◽  
pp. 989-1000 ◽  
Author(s):  
A. Maheshwari ◽  
P. Vellaisamy
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Short Range ◽  
Negative Binomial ◽  
Stationary Process ◽  
Long Range Dependence ◽  
Fractional Poisson Process ◽  
Poissonian Noise ◽  
Negative Binomial Process ◽  
Binomial Process

Abstract We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

Using EWMA control schemes for monitoring wafer quality in negative binomial process

2011 ◽  
Vol 51 (2) ◽  
pp. 400-405 ◽  
Author(s):  
Fong-Jung Yu ◽  
Yung-Yu Yang ◽  
Ming-Jaan Wang ◽  
Zhang Wu
Keyword(s):  
Negative Binomial ◽  
Control Schemes ◽  
Negative Binomial Process ◽  
Binomial Process

Negative binomial processes

1969 ◽  
Vol 6 (03) ◽  
pp. 633-647 ◽  
Author(s):  
Ole Barndorff-Nielsen ◽  
G. F. Yeo
Keyword(s):  
Finite Number ◽  
Compact Subset ◽  
Gaussian Processes ◽  
Negative Binomial ◽  
Sum Of Squares ◽  
Poisson Processes ◽  
Intensity Parameter ◽  
Negative Binomial Process ◽  
Binomial Process

Summary This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk , and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

scholarly journals On A Model for the Claim Number Process

1988 ◽  
Vol 18 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Matti Ruohonen
Keyword(s):  
Structure Function ◽  
Poisson Process ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Credibility Theory ◽  
Gamma Model ◽  
Function Fitting ◽  
Claim Number ◽  
Two Parameter

AbstractA model for the claim number process is considered. The claim number process is assumed to be a weighted Poisson process with a three-parameter gamma distribution as the structure function. Fitting of this model to several data encountered in the literature is considered, and the model is compared with the two-parameter gamma model giving the negative binomial distribution. Some credibility theory formulae are also presented.

scholarly journals L’exploration des ressources extractives non renouvelables : théorie économique, processus stochastique et vérification

2009 ◽  
Vol 53 (4) ◽  
pp. 559-586 ◽  
Author(s):  
Ph. Crabbé
Keyword(s):  
Economic Theory ◽  
Negative Binomial ◽  
Empirical Support ◽  
Turn Of The Century ◽  
Geological Information ◽  
Théorie Économique ◽  
Negative Binomial Process ◽  
The One ◽  
Binomial Process ◽  
Made In

AbstractMost of the literature devoted to the "theory of the mine" has been developed under certainty. It has been unable to explain the activity of exploration. The stochastic models of exploration were developed quasi-independently from economic theory. The purpose of this article is to survey both the mining and economic literature related to the "theory of the mine" under uncertainty and the exploration models since the turn of the century. The survey is complementary to the one made in this journal by F. Peterson and A.C. Fisher.The first part defines exploration as being essentially a search and information gathering activity. It reviews the contributions to the economic theory of exploration and resource stock uncertainty. It compares the optimal extraction path and the life cycle of the mine under stock uncertainty and stock certainty. It shows in particular that increasing the rate of discount is generally inappropriate to take account of stock uncertainty. Some partial equilibrium results on exploration are given as well. The presence of stock uncertainty or exploration in a general equilibrium model is shown to jeopardize the optimality of competitive allocations.The second part points out the wealth of the theoretical and empirical analysis of exploration as a stochastic process. It first reviews the literature on size distributions of reserves which gives strong theoretical and empirical support to the lognormal hypothesis. It then goes on to the exploration models which roughly speaking can be broken down into two groups. The Allais type models, better suited for relatively unexplored regions, which combine a Poisson or negative binomial process for discovery with a lognormal distribution for sizes. The Arps-Roberts-Kaufman type models, more adequate for "mature" regions, assume exhaustive sampling with probability proportional to size of discovery. Generally the treatment of the discovery process, to be distinguished from the sampling for sizes, and the handling of geological information are still woefully inadequate.The third and last part of the survey points out the gap which exists in the microeconomic literature about the study of random inputs. It suggests that the theory of dams and insurance and the theory of search especially adaptive search could be fruitfully used. Problems which remain to be tackled are the influence of stock uncertainty on grade of ore mined and on investment in capacity.

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