scholarly journals Mixed Compound Poisson Distributions

1986 ◽  
Vol 16 (S1) ◽  
pp. S59-S79 ◽  
Author(s):  
Gord Willmot
Keyword(s):  
Negative Binomial ◽  
Random Variable ◽  
Computational Formula ◽  
Inverse Gaussian ◽  
Poisson Distributions ◽  
Mixing Distributions ◽  
Chi Squared ◽  
General Distributions ◽  
Generalized Inverse Gaussian ◽  
Thick Tails

AbstractThe distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

scholarly journals Signal processing with Lévy information

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120433 ◽  
Author(s):  
Dorje C. Brody ◽  
Lane P. Hughston ◽  
Xun Yang
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Information Process ◽  
True Value ◽  
Information Processes ◽  
Noise Type ◽  
Exponential Moments

Lévy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Lévy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a ‘message’, then under the transformed measure the original Lévy process takes on the character of an ‘information process’. In this paper we develop a theory of such Lévy information processes. The underlying Lévy process, which we call the fiducial process, represents the ‘noise type’. Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.

scholarly journals The Mixed Bivariate Hofmann Distribution

2001 ◽  
Vol 31 (1) ◽  
pp. 123-138 ◽  
Author(s):  
J.F. Walhin ◽  
J. Paris
Keyword(s):  
Poisson Distribution ◽  
Gaussian Distribution ◽  
Negative Binomial ◽  
Inverse Gaussian Distribution ◽  
Inverse Gaussian ◽  
Bivariate Case ◽  
Univariate Case ◽  
Poisson Distributions ◽  
Aggregate Claims

AbstractIn this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case.We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature.This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.

scholarly journals Stein's method in two limit theorems involving the generalized inverse Gaussian distribution

2021 ◽  
Vol 16 (1) ◽  
pp. 2561-2586
Author(s):  
Essomanda Konzou
Keyword(s):  
Generalized Inverse ◽  
Rooted Tree ◽  
Random Variable ◽  
Stein’S Method ◽  
Stein's Method ◽  
Inverse Gaussian ◽  
Tree Distribution ◽  
Infinite Tree ◽  
Gig Distribution ◽  
Generalized Inverse Gaussian

The generalized hyperbolic (GH) distribution converges in law to the generalized inverse Gaussian (GIG) distribution under certain conditions on the parameters. When the edges of an infinite rooted tree are equipped with independent resistances that are inverse Gaussian or reciprocal inverse Gaussian distributions, the total resistance converges almost surely to some random variable which follows the reciprocal inverse Gaussian (RIG) distribution. In this paper we provide explicit upper bounds for the distributional distance between GH (resp. infinite tree) distribution and their limiting GIG (resp. RIG) distribution applying Stein's method.

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.

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