Precise large deviation results for the total claim amount under subexponential claim sizes

2008 ◽  
Vol 78 (10) ◽  
pp. 1206-1214 ◽  
Author(s):  
Aleksandras Baltrūnas ◽  
Remigijus Leipus ◽  
Jonas Šiaulys
Keyword(s):  
Large Deviation ◽  
Claim Amount ◽  
Total Claim Amount ◽  
Total Claim ◽  
Precise Large Deviation

scholarly journals Incorporating heterogeneity into the prediction of total claim amount

2017 ◽  
Vol 47 (4) ◽  
pp. 1-15
Author(s):  
Aslihan Senturk Acar ◽  
Ugur Karabey ◽  
Dario Gregori
Keyword(s):  
Claim Amount ◽  
Total Claim Amount ◽  
Total Claim

COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES

2015 ◽  
Vol 45 (3) ◽  
pp. 601-637 ◽  
Author(s):  
Raffaello Seri ◽  
Christine Choirat
Keyword(s):  
Random Variable ◽  
Cumulative Distribution ◽  
Claim Amount ◽  
Approximation Methods ◽  
Total Claim Amount ◽  
Aggregate Claim ◽  
Time Period ◽  
Total Claim ◽  
The Difference ◽  
Object Of Study

AbstractIn this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

scholarly journals A Functional Approach to Approximations for the Individual Risk Model

2004 ◽  
Vol 34 (2) ◽  
pp. 379-397 ◽  
Author(s):  
Susan M. Pitts
Keyword(s):  
Negative Binomial ◽  
Risk Model ◽  
Order Approximation ◽  
Functional Approach ◽  
Individual Risk ◽  
Claim Amount ◽  
Total Claim Amount ◽  
Total Claim ◽  
Binomial Approximation ◽  
The Individual

A functional approach is taken for the total claim amount distribution for the individual risk model. Various commonly used approximations for this distribution are considered, including the compound Poisson approximation, the compound binomial approximation, the compound negative binomial approximation and the normal approximation. These are shown to arise as zeroth order approximations in the functional set-up. By taking the derivative of the functional that maps the individual claim distributions onto the total claim amount distribution, new first order approximation formulae are obtained as refinements to the existing approximations. For particular choices of input, these new approximations are simple to calculate. Numerical examples, including the well-known Gerber portfolio, are considered. Corresponding approximations for stop-loss premiums are given.

scholarly journals EXPANSION IN BELL POLYNOMIALS OF THE DISTRIBUTION OF THE TOTAL CLAIM AMOUNT WITH WEIBULL-DISTRIBUTED CLAIM SIZES

2008 ◽  
Vol 49 (4) ◽  
pp. 495-501
Author(s):  
RAMON LACAYO
Keyword(s):  
Random Number ◽  
Negative Binomial ◽  
Claim Amount ◽  
Bell Polynomials ◽  
Total Claim Amount ◽  
Binomial Distributions ◽  
Probabilistic Distribution ◽  
Satisfactory Model ◽  
Total Claim ◽  
Fixed Period

AbstractThe total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.

scholarly journals Prediction of Outstanding Liabilities in Non-Life Insurance

1993 ◽  
Vol 23 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Ragnar Norberg
Keyword(s):  
Life Insurance ◽  
Linear Prediction ◽  
Basic Result ◽  
Claim Amount ◽  
Insurance Company ◽  
Life Insurance Company ◽  
Total Claim Amount ◽  
Generalized Poisson ◽  
Total Claim ◽  
The Individual

AbstractA fully time-continuous approach is taken to the problem of predicting the total liability of a non-life insurance company. Claims are assumed to be generated by a non-homogeneous marked Poisson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Poisson processes. By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner. Special attention is given to the case where the claim intensity per risk unit is a stationary stochastic process. A theory of continuous linear prediction is instrumental.

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