Modeling of Motor Insurance Extreme Claims through Appropriate Statistical Distributions

In the area of insurance, probability modeling has a wide variety of applications. In life insurance, the compensation sum is calculated in advance and may often be estimated using actuarial techniques, while in motor insurance, the claim amount is generally not known in advance. In the insurance business, the improvement of actuarial risk control strategies is an essential technique for controlling insurance risk. Although an insurance company’s risk assessment about its solvency is a complex and detailed problem, its solution begins with statistical modeling of individual claims’ amounts. This article emphasizes the possible ways of obtaining a suitable probability distribution model that accurately explains insurance risks and how to use such a model for risk management purposes. For this reason, we have applied modern programming techniques and statistical software implemented the methods provided based on data on premium amounts of third-party motor insurance claims.

On transform orders for largest claim amounts

This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

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

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.

Risk Analysis of the Copula Dependent Aggregate Discounted Claims with Weibull Inter-Arrival Time

We model the recursive moments of aggregate discounted claims, assuming the inter-claim arrival time follows a Weibull distribution to accommodate overdispersed and underdispersed data set. We use a copula to represent the dependence structure between the inter-claim arrival time and its subsequent claim amount. We then use the Laplace inversion via the Gaver-Stehfest algorithm to solve numerically the first and second moments, which takes the form of a Volterra integral equation (VIE). We compute the average and variance of the aggregate discounted claims under the Farlie-Gumbel-Morgenstern (FGM) copula and conduct a sensitivity analysis under various Weibull inter-claim parameters and claim-size parameters. The comparison between the equidispersed, overdispersed and underdispersed counting processes shows that when claims arrive at times that vary more than is expected, insured lives can expect to pay higher premium, and vice versa for the case of claims arriving at times that vary less than expected. Upon comparing the Weibull risk process with an equivalent Poisson process, we also found that copulas with a wider range of dependency parameter such as the Frank and Heavy Right Tail (HRT), have a greater impact on the value of moments as opposed to modeling under FGM copula with weak dependence structure.

Ordering results for smallest claim amounts from two portfolios of risks with dependent heterogeneous exponentiated location-scale claims

Let $\{Y_{1},\ldots ,Y_{n}\}$ be a collection of interdependent nonnegative random variables, with $Y_{i}$ having an exponentiated location-scale model with location parameter $\mu _i$ , scale parameter $\delta _i$ and shape (skewness) parameter $\beta _i$ , for $i\in \mathbb {I}_{n}=\{1,\ldots ,n\}$ . Furthermore, let $\{L_1^{*},\ldots ,L_n^{*}\}$ be a set of independent Bernoulli random variables, independently of $Y_{i}$ 's, with $E(L_{i}^{*})=p_{i}^{*}$ , for $i\in \mathbb {I}_{n}.$ Under this setup, the portfolio of risks is the collection $\{T_{1}^{*}=L_{1}^{*}Y_{1},\ldots ,T_{n}^{*}=L_{n}^{*}Y_{n}\}$ , wherein $T_{i}^{*}=L_{i}^{*}Y_{i}$ represents the $i$ th claim amount. This article then presents several sufficient conditions, under which the smallest claim amounts are compared in terms of the usual stochastic and hazard rate orders. The comparison results are obtained when the dependence structure among the claim severities are modeled by (i) an Archimedean survival copula and (ii) a general survival copula. Several examples are also presented to illustrate the established results.

Conditional mean risk sharing in the individual model with graphical dependencies

Conditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

A Study on Days of hospitalization of insured with the claims data

: The health insurance sector has grown at a double digit growth rate in India in the past decade. The Government schemes for the individuals' insurance coverage from the low-income group have resulted in higher penetration. Raise in the disposable income of the middle income and awareness of health insurance has led to self-subscription by private individuals. An increase in insurance penetration has also led to an increment in the claims for the insurer. However, the insurance coverage has been actively subscribed by the population in the age group of 19-64 years. The average claim amount processed is higher in infants and aged people. Days of hospitalization for the insured treatment help the insurers derive the claims' amounts and hence can budget the reserves accordingly. Days of hospitalization is found to have a strong positive relationship with age. Logistic regression of Half-yearly bins data has the predictability of Days of hospitalization of claimants at 69%.

Claim Amount Forecasting and Pricing of Automobile Insurance Based on the BP Neural Network

The BP neural network model is a hot issue in recent academic research, and it has been successfully applied to many other fields, but few researchers apply the BP neural network model to the field of automobile insurance. The main method that has been used in the prediction of the total claim amount in automobile insurance is the generalized linear model, where the BP neural network model could provide a different approach to estimate the total claim loss. This paper uses a genetic algorithm to optimize the structure of the BP neural network at first, and the calculation speed is significantly improved. At the same time, by considering the overfitting problem, an early stop method is introduced to avoid the overfitting problem. In the model, a three-layer BP neural network model, which includes the input layer, hidden layer, and output layer, is trained. With consideration of various factors, a total claim amount prediction model is established, and the trained BP neural network model is used to predict the total claim amount of automobile insurance based on the data of the training set. The results show that the accuracy of the prediction by using the BP neural network model to both the data of Shandong Province and to the data of six cities is over 95%. Then, the predicted total claim amount is used to calculate premiums for five cities in Shandong Province according to credibility theory. The results show that the average premium of the five cities is slightly higher than the actual claim amount of the city. The combination of BP neural network and credibility theory can perform accurate claim amount estimation and pricing for automobile insurance, which can effectively improve the current situation of the automobile insurance business and promote the development of insurance industry.

An Expectation-Maximization Algorithm for the Exponential-Generalized Inverse Gaussian Regression Model with Varying Dispersion and Shape for Modelling the Aggregate Claim Amount

This article presents the Exponential–Generalized Inverse Gaussian regression model with varying dispersion and shape. The EGIG is a general distribution family which, under the adopted modelling framework, can provide the appropriate level of flexibility to fit moderate costs with high frequencies and heavy-tailed claim sizes, as they both represent significant proportions of the total loss in non-life insurance. The model’s implementation is illustrated by a real data application which involves fitting claim size data from a European motor insurer. The maximum likelihood estimation of the model parameters is achieved through a novel Expectation Maximization (EM)-type algorithm that is computationally tractable and is demonstrated to perform satisfactorily.

Some Remarks on Kant's Theory of Experience (Translated by M. Evstigneev)

Kant never tires of telling us that Nature and the Space and Time which are its forms exist as a system of “representations.” Now a prepresentation is either a representing or a something represented. Does Kant mean that nature is a system of representings? Or that it is a system of representeds? And, in any case, what would the claim amount to? (Sellars W. (1974) Some Remarks on Kant’s Theory of Experience. In: Essays in Philosophy and Its History. Philosophical Studies Series in Philosophy, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2291-0_3)