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.

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.

Poisson size-biased Lindley distribution and its applications

2021 ◽  
Author(s):  
Showkat Ahmad Dar ◽  
Anwar Hassan ◽  
Peer Bilal Ahmad
Keyword(s):  
Parameter Estimation ◽  
Count Data ◽  
Hazard Rate ◽  
Risk Model ◽  
Probability Generating Function ◽  
Likelihood Estimation ◽  
Lindley Distribution ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Erlang Distributions

In this paper, a new model for count data is introduced by compounding the Poisson distribution with size-biased three-parameter Lindley distribution. Statistical properties, such as reliability, hazard rate, reverse hazard rate, Mills ratio, moments, shewness, kurtosis, moment genrating function, probability generating function and order statistics, have been discussed. Moreover, the collective risk model is discussed by considering the proposed distrubution as the primary distribution and the expoential and Erlang distributions as the secondary ones. Parameter estimation is done using maximum likelihood estimation (MLE). Finally a real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in modeling count dataset.

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 Posterior Regret Γ-Minimax Estimation of Insurance Premium in Collective Risk Model

2008 ◽  
Vol 38 (1) ◽  
pp. 277-291 ◽  
Author(s):  
Agata Boratyńska
Keyword(s):  
Loss Function ◽  
Unknown Parameter ◽  
Risk Model ◽  
Insurance Premium ◽  
Loss Functions ◽  
Expected Number ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Asymmetric Loss Function ◽  
Robust Procedures

The collective risk model for the insurance claims is considered. The objective is to estimate a premium which is defined as a functional H specified up to an unknown parameter θ (the expected number of claims). Four principles of calculating a premium are applied. The Bayesian methodology, which combines the prior knowledge about a parameter θ with the knowledge in the form of a random sample is adopted. Two loss functions (the square-error loss function and the asymmetric loss function LINEX) are considered. Some uncertainty about a prior is assumed by introducing classes of priors. Considering one of the concepts of robust procedures the posterior regret Γ-minimax premiums are calculated, as an optimal robust premiums. A numerical example is presented.

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