Matrix-Form Recursions for a Family of Compound Distributions

2010 ◽  
Vol 40 (1) ◽  
pp. 351-368 ◽  
Author(s):  
Xueyuan Wu ◽  
Shuanming Li
Keyword(s):  
Risk Model ◽  
Discrete Distribution ◽  
Recursive Formula ◽  
Matrix Form ◽  
Claim Amount ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Compound Distribution ◽  
Phase Type ◽  
Phase Type Distributions

AbstractIn this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

2011 ◽  
Vol 5 (2) ◽  
pp. 163-179 ◽  
Author(s):  
Kok Keng Siaw ◽  
Xueyuan Wu ◽  
David Pitt ◽  
Yan Wang
Keyword(s):  
Risk Model ◽  
Likelihood Estimation ◽  
Real Data ◽  
Recursive Formula ◽  
Matrix Form ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Potential Applications ◽  
Aggregate Claims ◽  
Recursive Evaluation

AbstractThis paper aims to evaluate the aggregate claims distribution under the collective risk model when the number of claims follows a so-called generalised (a, b, 1) family distribution. The definition of the generalised (a, b, 1) family of distributions is given first, then a simple matrix-form recursion for the compound generalised (a, b, 1) distributions is derived to calculate the aggregate claims distribution with discrete non-negative individual claims. Continuous individual claims are discussed as well and an integral equation of the aggregate claims distribution is developed. Moreover, a recursive formula for calculating the moments of aggregate claims is also obtained in this paper. With the recursive calculation framework being established, members that belong to the generalised (a, b, 1) family are discussed. As an illustration of potential applications of the proposed generalised (a, b, 1) distribution family on modelling insurance claim numbers, two numerical examples are given. The first example illustrates the calculation of the aggregate claims distribution using a matrix-form Poisson for claim frequency with logarithmic claim sizes. The second example is based on real data and illustrates maximum likelihood estimation for a set of distributions in the generalised (a, b, 1) family.

The individual risk model: a compound distribution

1989 ◽  
Vol 116 (1) ◽  
pp. 101-107 ◽  
Author(s):  
R. J. Verrall
Keyword(s):  
Risk Model ◽  
Individual Risk ◽  
Risk Models ◽  
Collective Models ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Compound Distribution ◽  
Compound Binomial ◽  
Mean And Variance ◽  
The Individual

The two approaches to modelling aggregate claims—the individual and the collective models—have been regarded as arising by considering a portfolio of policies in different ways. The individual risk model (IRM) is derived by considering the claims on individual policies and summing over all policies in the portfolio, while the collective risk model (CRM) is derived from the portfolio as a whole. This is sometimes held to be the main difference between the IRM and the CRM. In fact the IRM can be derived in exactly the same way as the CRM and can be regarded as a compound binomial distribution. This makes a unified treatment of risk models possible, simplifies the calculation of the mean and variance of the IRM, and facilitates the calculation of higher moments.

scholarly journals A Desirable Aspect in the Variance Premium in a Collective Risk Model

2020 ◽  
Vol 29 (1) ◽  
pp. 395
Author(s):  
Agustín Hernández-Bastida ◽  
Mª Del Pilar Fernández-Sánchez ◽  
Emilio Gómez-Déniz
Keyword(s):  
Prior Distribution ◽  
Risk Model ◽  
Structure Functions ◽  
Exponential Model ◽  
Claim Amount ◽  
Collective Risk Model ◽  
Collective Risk

This paper focuses on the study of the Collective and Bayes Premiums, under the Variance Premium Principle, in the classic Collective Risk Poisson-Exponential Model. A bivariate prior distribution is considered for both the parameter of the distribution of the number of claims and that of the distribution of the claim amount, assuming independence between these parameters. Furthermore, we analyze the consequences on these premiums of small levels of contamination in the structure functions, and find that the premiums are not sensitive to small levels of uncertainty. These results extend the conclusions obtained in Gómez-Déniz et al. (2000), where only variations in the parameter for the number of claims and its effects on premiums were studied.

ON THE FAMILY OF DISTRIBUTIONS WITH ACTUARIAL APPLICATIONS

2021 ◽  
pp. 1-26
Author(s):  
Deepesh Bhati ◽  
Enrique Calderín-Ojeda
Keyword(s):  
Negative Binomial ◽  
Risk Model ◽  
Recursive Formula ◽  
Bell Polynomials ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Insurance Portfolio ◽  
The Family ◽  
Out Of Sample ◽  
Family Of Distributions

ABSTRACT In this paper, a new three-parameter discrete family of distributions, the $$r{\cal B}ell$$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

scholarly journals A Multi-Year Microlevel Collective Risk Model

2021 ◽  
Author(s):  
Rosy Oh ◽  
Himchan Jeong ◽  
Jae Youn Ahn ◽  
Emiliano A. Valdez
Keyword(s):  
Risk Model ◽  
Collective Risk Model ◽  
Collective Risk

scholarly journals Estimation of aggregate losses of secondary cancer cases using PH Panjer class (a,b,1) distributions,

2021 ◽  
Vol 16 (4) ◽  
pp. 3941-3959
Author(s):  
Cynthia Mwende Mwau ◽  
Patrick Guge Weke ◽  
Bundi Davis Ntwiga ◽  
Joseph Makoteku Ottieno
Keyword(s):  
Transition Probabilities ◽  
Generalized Pareto Distribution ◽  
Kolmogorov Equation ◽  
Claim Amount ◽  
Secondary Cancer ◽  
Class A ◽  
Generalized Pareto ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Aggregate Loss

This research in-cooperates phase type distributions of Panjer class \((a,b,1)\) in estimation of aggregate loss probabilities of secondary cancer. Matrices of the phase type distributions are derived using Chapman-Kolmogorov equation and transition probabilities estimated using modified Kaplan-Meir and consequently the transition intensities and transition probabilities. Stationary probabilities of the Markov chains represents $\vec{\gamma}$. Claim amount are modeled using OPPL, TPPL, Pareto, Generalized Pareto and Wei-bull distributions. PH ZT Poisson with Generalized Pareto distribution provided the best fit.

scholarly journals Collective risk model in non-life insurance

2013 ◽  
Vol 15 (2) ◽  
pp. 163-172
Author(s):  
Zlata Djuric
Keyword(s):  
Life Insurance ◽  
Risk Model ◽  
Collective Risk Model ◽  
Collective Risk
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