Assessing the Performance of the Discrete Generalised Pareto Distribution in Modelling Non-Life Insurance Claims

The generalised Pareto distribution (GPD) offers a family of probability spaces which support threshold exceedances and is thus suitable for modelling high-end actuarial risks. Nonetheless, its distributional continuity presents a critical limitation in characterising data of discrete forms. Discretising the GPD, therefore, yields a derived distribution which accommodates the count data while maintaining the essential tail modelling properties of the GPD. In this paper, we model non-life insurance claims under the three-parameter discrete generalised Pareto (DGP) distribution. Data for the study on reported and settled claims, spanning the period 2012–2016, were obtained from the National Insurance Commission, Ghana. The maximum likelihood estimation (MLE) principle was adopted in fitting the DGP to yearly and aggregated data. The estimation involved two steps. First, we propose a modification to the μ and μ + 1 frequency method in the literature. The proposal provides an alternative routine for generating initial estimators for MLE, in cases of varied count intervals, as is a characteristic of the claim data under study. Second, a bootstrap algorithm is implemented to obtain standard errors of estimators of the DGP parameters. The performance of the DGP is compared to the negative binomial distribution in modelling the claim data using the Akaike and Bayesian information criteria. The results show that the DGP is appropriate for modelling the count of non-life insurance claims and provides a better fit to the regulatory claim data considered.

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Modeling of Motor Vehicle Claims Using Extreme Value Methodology and Monte Carlo Stimulation: A Comparative Study

Asia Pacific Business Review ◽

10.1177/097324700800400409 ◽

2008 ◽

Vol 4 (4) ◽

pp. 96-103

Author(s):

S. Chandrasekhar

Keyword(s):

Monte Carlo ◽

Pareto Distribution ◽

Motor Vehicle ◽

Claim Data ◽

Extreme Value ◽

Extreme Value Analysis ◽

Claim Amount ◽

Insurance Claims ◽

Value Analysis ◽

Generalised Pareto Distribution

Motor Vehicle Insurance claims form a substantial component of Non life insurance claims and it is also growing with increasing number of vehicles on roads. It is also desirable to have an idea of what will be the likely claim amount for the coming future (Monthly, Quarterly, Yearly) based on past claim data. If one looks at the claim amount one can make out that there will be few large claims compared to large number of average and below average claims. Thus the distributions of claims do not follow a Symmetric pattern which makes it difficult using normal Statistical analysis. The methodology followed to analyze such data is known as Extreme value Analysis. Extreme value analysis is a general name which covers (i) Generalised Extreme Value (GEV) (ii) Generalised Pareto Distribution (GPD). Basically these techniques can deal with non symmetric shape of the distribution which is close to reality. Normally one fits a generalised Extreme Value distribution (GEV)/Generalised Pareto Distribution (GPD) and using parameters of fitted distribution future, forecast of likely losses can be predicted. Second method of analyzing such data is using methodology of simulation. Here we fit a Poisson distribution for arrival of claims and weibull/pareto/Lognormal for claim amount. Using Monte Carlo Simulation one combines both the distributions for future prediction of claim amount. This paper shows a comparison of the above techniques on motor vehicle claims data.

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Survival Probabilities Based on Pareto Claim Distributions

Astin Bulletin ◽

10.1017/s0515036100006620 ◽

1980 ◽

Vol 11 (1) ◽

pp. 61-71 ◽

Cited By ~ 20

Author(s):

Hilary L. Seal

Keyword(s):

Characteristic Function ◽

Life Insurance ◽

Hypergeometric Functions ◽

Functional Form ◽

Pareto Distribution ◽

Claim Data ◽

Automobile Insurance ◽

Numerical Work ◽

Confluent Hypergeometric Functions ◽

Image Position

It is commonly thought that the characteristic function (Fourier transform) of the Pareto distribution has no known functional form (e.g. Seal, 1978, pp. 14, 40, 57). This is quite untrue. Nevertheless the characteristic function of the Pareto density is conspicuously absent from standard reference works even when the Pareto distribution itself receives substantial comment (e.g. Haight, 1961; Johnson and Kotz, 1970, Ch. 19; Patel, Kapadia and Owen, 1976, § 1. 5).The Pareto density may be writtenwith distribution functionmean = p/(v − 1) and variance = b2v(v − 1)2(v−2). These are infinite when v≤1 and v≤2, respectively. Its Laplace transform (s= c + iu)where E is the generalized exponential integral (Pagurova, 1961) and can be written in terms of incomplete gamma or confluent hypergeometric functions (Slater, 1960, Sec. 5.6). When s = − it β(s) becomes the characteristic function (see Appendix I).As Benktander (1970) tells us, the Pareto distribution has been particularly successful at representing the distribution of the larger claim amounts. In earlier years it was employed to represent the distribution of life insurance sums assured but more recently it has been used for the claim distributions of fire and automobile insurance. Table 1 provides the v-values we have been able to locate. Note that the variance of the distribution is infinite when v≤2 and if it were not for the anomalous v-values of Andersson (1971) we would have ventured the opinion that modern claim data encourage the assumption that v>2. In our numerical work we have used v = 2.7 and smaller values might change some of the computer rules we have proposed in Appendix II.

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On Modelling Extreme Damages from Natural Disasters in Kenya

10.5772/intechopen.94578 ◽

2021 ◽

Author(s):

Carolyne Ogutu ◽

Antony Rono

Keyword(s):

Natural Disasters ◽

Negative Binomial ◽

Extreme Value Distribution ◽

Likelihood Estimation ◽

High Threshold ◽

Frequency Of Occurrence ◽

Probability Weighted Moments ◽

Log Likelihood ◽

Future Events ◽

Generalised Pareto Distribution

We seek to develop a distribution to model the extreme damages resulting from Natural Disasters in Kenya.The distribution is based on the Compound Extreme Value Distribution, which takes into account both the distributions of the frequency of occurrence and magnitude of the events. Threshold modelling is employed, where the extreme damages are identified as the points that lie above a sufficiently high threshold. The distribution of the number of the exceedance is found to be Negative Binomial, while that of the severity is approximated by a Generalised Pareto Distribution. Maximum likelihood estimation is used to estimate the parameters, and the log-likelihood is maximised using numerical methods. Probability weighted moments estimation is used to determine the starting values for the iterations. Prediction study is then carried out to investigate the performance of the proposed distribution in predicting future events.

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Estimating the Cost of Car Warranty in Indonesia using the Gertsbakh-Kordonsky Method

InPrime: Indonesian Journal of Pure and Applied Mathematics ◽

10.15408/inprime.v2i1.14556 ◽

2020 ◽

Vol 2 (1) ◽

pp. 33-40

Author(s):

Anggis Sagitarisman ◽

Aceng Komarudin Mutaqin

Keyword(s):

Maximum Likelihood ◽

Likelihood Estimation ◽

Type A ◽

Claim Data ◽

Likelihood Method ◽

Selling Price ◽

One Dimensional ◽

Warranty Period ◽

Kolmogorov Smirnov ◽

Warranty Costs

AbstractCar manufacturers in Indonesia need to determine reasonable warranty costs that do not burden companies or consumers. Several statistical approaches have been developed to analyze warranty costs. One of them is the Gertsbakh-Kordonsky method which reduces the two-dimensional warranty problem to one dimensional. In this research, we apply the Gertsbakh-Kordonsky method to estimate the warranty cost for car type A in XYZ company. The one-dimensional data will be tested using the Kolmogorov-Smirnov to determine its distribution and the parameter of distribution will be estimated using the maximum likelihood method. There are three approaches to estimate the parameter of the distribution. The difference between these three approaches is in the calculation of mileage for units that do not claim within the warranty period. In the application, we use claim data for the car type A. The data exploration indicates the failure of car type A is mostly due to the age of the vehicle. The Kolmogorov-Smirnov shows that the most appropriate distribution for the claim data is the three-parameter Weibull. Meanwhile, the estimated using the Gertsbakh-Kordonsky method shows that the warranty costs for car type A are around 3.54% from the selling price of this car unit without warranty i.e. around Rp. 4,248,000 per unit.Keywords: warranty costs; the Gertsbakh-Kordonsky method; maximum likelihood estimation; Kolmogorov-Smirnov test. AbstrakPerusahaan produsen mobil di Indonesia perlu menentukan biaya garansi yang bersifat wajar tidak memberatkan perusahaan maupun konsumen. Beberapa pendekatan statistik telah dikembangkan untuk menganalisis biaya garansi. Salah satunya adalah metode Gertsbakh-Kordonsky yang mereduksi masalah garansi dua dimensi menjadi satu dimensi. Pada penelitian ini, metode Gertsbakh-Kordonsky akan digunakan untuk mengestimasi biaya garansi untuk mobil tipe A pada perusahaan XYZ. Data satu dimensi hasil reduksi diuji kecocokan distribusinya menggunakan uji kecocokan Kolmogorov-Smirnov dan taksiran parameter distribusinya menggunakan metode penaksir kemungkinan maksimum. Ada tiga pendekatan yang digunakan untuk menaksir parameter distribusi. Perbedaan dari ketiga pendekatan tersebut terletak pada perhitungan jarak tempuh untuk unit yang tidak melakukan klaim dalam periode garansi. Sebagai bahan aplikasi, kami menggunakan data klaim unit mobil tipe A. Hasil eksplorasi data menunjukkan bahwa kegagalan mobil tipe A lebih banyak disebabkan karena faktor usia kendaraan. Hasil uji kecocokan distribusi untuk data hasil reduksi menunjukkan bahwa distribusi yang cocok adalah distribusi Weibull 3-parameter. Sementara itu, hasil perhitungan taksiran biaya garansi menunjukan bahwa taksiran biaya garansi untuk unit mobil tipe A sekitar 3,54% dari harga jual unit mobil tipe A tanpa garansi, atau sekitar Rp. 4.248.000,- per unit.Kata Kunci: biaya garansi; metode Gertsbakh-Kordonsky; penaksiran kemungkinan maksimum; uji Kolmogorov-Smirnov.

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Uncertainties in return values from extreme value analysis of peaks over threshold using the generalised Pareto distribution

Ocean Engineering ◽

10.1016/j.oceaneng.2020.107725 ◽

2021 ◽

pp. 107725

Author(s):

Philip Jonathan ◽

David Randell ◽

Jenny Wadsworth ◽

Jonathan Tawn

Keyword(s):

Pareto Distribution ◽

Extreme Value ◽

Extreme Value Analysis ◽

Peaks Over Threshold ◽

Value Analysis ◽

Generalised Pareto Distribution

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The model of life insurance claims with actuarial smoothing approach by using GLM Poisson regression

10.1063/5.0032179 ◽

2020 ◽

Author(s):

Ully Saputri ◽

Dodi Devianto

Keyword(s):

Life Insurance ◽

Poisson Regression ◽

Insurance Claims

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Identifying Stabilising Effects on Survey Based Life Satisfaction Using Quasi-maximum Likelihood Estimation

Journal of Happiness Studies ◽

10.1007/s10902-021-00376-3 ◽

2021 ◽

Author(s):

Johannes Klement

Keyword(s):

Life Satisfaction ◽

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Negative Binomial ◽

Measurement Instrument ◽

Likelihood Estimation ◽

Satisfaction Scale ◽

Quasi Maximum Likelihood ◽

The Stability

AbstractTo which extent do happiness correlates contribute to the stability of life satisfaction? Which method is appropriate to provide a conclusive answer to this question? Based on life satisfaction data of the German SOEP, we show that by Negative Binomial quasi-maximum likelihood estimation statements can be made as to how far correlates of happiness contribute to the stabilisation of life satisfaction. The results show that happiness correlates which are generally associated with a positive change in life satisfaction, also stabilise life satisfaction and destabilise dissatisfaction with life. In such as they lower the probability of leaving positive states of life satisfaction and increase the probability of leaving dissatisfied states. This in particular applies to regular exercise, volunteering and living in a marriage. We further conclude that both patterns in response behaviour and the quality of the measurement instrument, the life satisfaction scale, have a significant effect on the variation and stability of reported life satisfaction.

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