Poisson-exponential Model Research Articles

Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth.

A goodness-of-fit test for the compound Poisson exponential model

On the basis of bivariate data, assumed to be observations of independent copies of a random vector (S,N), we consider testing the hypothesis that the distribution of (S,N) belongs to the parametric class of distributions that arise with the compound Poisson exponential model. Typically, this model is used in stochastic hydrology, with N as the number of raindays, and S as total rainfall amount during a certain time period, or in actuarial science, with N as the number of losses, and S as total loss expenditure during a certain time period. The compound Poisson exponential model is characterized in the way that a specific transform associated with the distribution of (S,N) satisfies a certain differential equation. Mimicking the function part of this equation by substituting the empirical counterparts of the transform we obtain an expression the weighted integral of the square of which is used as test statistic. We deal with two variants of the latter, one of which being invariant under scale transformations of the S-part by fixed positive constants. Critical values are obtained by using a parametric bootstrap procedure. The asymptotic behavior of the tests is discussed. A simulation study demonstrates the performance of the tests in the finite sample case. The procedure is applied to rainfall data and to an actuarial dataset. A multivariate extension is also discussed.

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

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.

MODEL EKSPONENSIAL GANDA PADA PROSES STOKASTIK (STUDI KASUS DI STASIUN PURWOSARI)

In general, mathematical modeling is divided into two, namely the model of deterministic and stochastic models. On stochastic modeling involves several processes among them are the Poisson process, the process of Bernoulli, Gaussian processes, the process of renewal and other processes. Specifically for the Poisson process often found in modeling queuing theory. At Poisson process there are four kinds of sub model that can be formed that is Double Poisson models, Exponential Poisson models, Poisson Exponential model, and Double Exponential models. In this paper will discuss the Double Exponential model in stochastic processes , specifically for the Poisson process. Analysis was performed on the data arrival time and service time. The model is a model (M / M / c) : ( GD / ~, ~) which is a double exponential model in stochastic processes. Keywords: Double Exponential, Poisson Process, Stochastic Process

The Poisson-exponential model for recurrent event data: an application to bowel motility data

This paper presents a new parametric model for recurrent events, in which the time of each recurrence is associated to one or multiple latent causes and no information is provided about the responsible cause for the event. This model is characterized by a rate function and it is based on the Poisson-exponential distribution, namely the distribution of the maximum among a random number (truncated Poisson distributed) of exponential times. The time of each recurrence is then given by the maximum lifetime value among all latent causes. Inference is based on a maximum likelihood approach. A simulation study is performed in order to observe the frequentist properties of the estimation procedure for small and moderate sample sizes. We also investigated likelihood-based tests procedures. A real example from a gastroenterology study concerning small bowel motility during fasting state is used to illustrate the methodology. Finally, we apply the proposed model to a real data set and compare it with the classical Homogeneous Poisson model, which is a particular case.

On Infinitely Divisible Exponential Dispersion Model Related to Poisson-Exponential Distribution

We construct a univariate exponential dispersion model comprised of discrete infinitely divisible distributions. This model emerges in the theory of branching processes. We obtain a representation for the Lévy measure of relevant distributions and characterize their laws as Poisson mixtures and/or compound Poisson distributions. The regularity of the unit variance function of this model is employed for the derivation of approximations by the Poisson-exponential model. We emphasize the role of the latter class. We construct local approximations relating them to properties of special functions and branching diffusions.

A technique for selecting the threshold in applications of the poisson-exponential flood model

A problem usually encountered when modeling partial duration flood series with the Poisson-exponential model is the difficulty in finding a threshold or base level over which the model is applicable. This threshold must define a flood series such that the number of events per time period is Poisson-distributed, the events are mutually independent, and their magnitudes are exponentially distributed. This article describes and uses empirical techniques designed for dealing with such a problem. These techniques make use of the probability-generating function for testing the assumption of Poisson-distributed flood count data and a test based on the standardized P-P probability plot for the exponential distribution. Examples using these simple and graphical approaches are presented and discussed. Results show that these techniques are very promising among mathematically based methods for threshold selection in partial duration series analysis.

On the relationship between the Poisson-exponential model and the non-central chi-squared distribution

Abstract Two comments are made on the relationship between the Poisson-exponential model of insurance claims and the non-central chi-squared distribution. The first relates the Poisson-exponential to the recently introduced noncentral chi-squared on zero degrees-of-freedom; the second expands on the link between that distribution and the non-central chi-squared distribution on four degrees-of-freedom, noted by Boyle (1978).

A remark on “The Poisson-Exponential Model and the Non-Central Chi-squared Distribution”

Abstract In the following it is proved, that the simulation technique described by Boyle (1978) is not correct and according to that, the conclusions based on the results of the simulations are of limited value.

The poisson-exponential model and the Non-Central Chi-squared distribution

Summary A connection is established between the Poisson-Exponential model and the Non-central Chi-squared distribution. This provides an alternative numerical method of calculating the distribution of aggregate claims and the distribution of the stop-loss reinsurance premium. A connection between this model and some recent work in the theory of option pricing is pointed out.