A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.     

Modelling claim number using a new mixture model: negative binomial gamma distribution

ABSTRACT

In actuarial applications, mixed Poisson distributions are widely used for modelling claim counts as observed data on the number of claims often exhibit a variance noticeably exceeding the mean. In this study, a new claim number distribution is obtained by mixing negative binomial parameter p which is reparameterized as p?=?exp( ?λ) with Gamma distribution. Basic properties of this new distribution are given. Maximum likelihood estimators of the parameters are calculated using the Newton–Raphson and genetic algorithm (GA). We compared the performance of these methods in terms of efficiency by simulation. A numerical example is provided.     

Risk aggregation with dependence and overdispersion based on the compound Poisson INAR(1) process

Abstract

This paper considers an extension of the classical discrete time risk model for which the claim numbers are assumed to be temporal dependence and overdispersion. The risk model proposed is based on the first-order integer-valued autoregressive (INAR(1)) process with discrete compound Poisson distributed innovations. The explicit expression for the moment generating function of the discounted aggregate claim amount is derived. Some numerical examples are provided to illustrate the impacts of dependence and overdispersion on related quantities such as the stop-loss premium, the value at risk and the tail value at risk.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

The Gerber-Shiu function for the compound Poisson Omega model with a three-step premium rate

Abstract

The compound Poisson Omega model is considered in the presence of a three-step premium rate. Firstly, the integral equations and the integro-differential equations for the Gerber-Shiu expected discounted penalty function are derived. Secondly, the integro-differential equations for the Gerber-Shiu expected discounted penalty function are determined in three different initial conditions. The results are then used to find the bankruptcy probability. Finally, the special cases where the claim size distribution is exponential be discussed in some detail in order to illustrate the effect of the model with three-step premium rate.     

Hidden Markov model for discrete circular–linear wind data time series

ABSTRACT

In this work, we deal with a bivariate time series of wind speed and direction. Our observed data have peculiar features, such as informative missing values, non-reliable measures under a specific condition and interval-censored data, that we take into account in the model specification. We analyse the time series with a non-parametric Bayesian hidden Markov model, introducing a new emission distribution, suitable to model our data, based on the invariant wrapped Poisson, the Poisson and the hurdle density. The model is estimated on simulated datasets and on the real data example that motivated this work.     

The ARL of modified Shewhart control charts for conditionally heteroskedastic models

In this article we consider the modified Shewhart control chart for ARCH processes and introduce it for threshold ARCH (TARCH) ones. For both charts, we determine bounds for the distribution of the in-control run length (RL) and, consequently, for its average (ARL), both depending only on the distribution of the generating white noise, the model parameters and the critical value. For the ARCH model, we compare our bounds with others available in literature and show how they improve the existing ones. We present a simulation study to assess the quality of the bounds calculated for the ARL.     

The discrete Lindley distribution: properties and applications

Modelling count data is one of the most important issues in statistical research. In this paper, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution. The model obtained is over-dispersed and competitive with the Poisson distribution to fit automobile claim frequency data. After revising some of its properties a compound discrete Lindley distribution is obtained in closed form. This model is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model. The new compound distribution fades away to zero much more slowly than the classical compound Poisson distribution, being therefore suitable for modelling extreme data.     

Bayesian robustness of the compound Poisson distribution under bidimensional prior: an application to the collective risk model

The distribution of the aggregate claims in one year plays an important role in Actuarial Statistics for computing, for example, insurance premiums when both the number and size of the claims must be implemented into the model. When the number of claims follows a Poisson distribution the aggregated distribution is called the compound Poisson distribution. In this article we assume that the claim size follows an exponential distribution and later we make an extensive study of this model by assuming a bidimensional prior distribution for the parameters of the Poisson and exponential distribution with marginal gamma. This study carries us to obtain expressions for net premiums, marginal and posterior distributions in terms of some well-known special functions used in statistics. Later, a Bayesian robustness study of this model is made. Bayesian robustness on bidimensional models was deeply treated in the 1990s, producing numerous results, but few applications dealing with this problem can be found in the literature.     

Prediction in a mixed Poisson cluster model

Motivated by insurance applications, a mixed Poisson cluster model is considered, where the cluster center process is a mixed Poisson process and descendant processes are additive processes. Each point of the center process represents a claim’s reported time and descendant processes are interpreted as processes of the corresponding payments or number of payments. In this study, we focus on the process aggregating all separate claim’s payment processes. Given the past observations, we study prediction of future increments and their mean-squared errors, also revealing the dependency between future increments from non-reported (IBNR) claims and the past available information. In the existing literature, they are independent since models were considered with a purely Poissonian center process. We derive computationally reasonable expressions for predictors and their variances.     

Marginal-sum and maximum-likelihood estimation in a multiplicative tariff

It is well known that, for a multiplicative tariff with independent Poisson distributed claim numbers in the different tariff cells, the maximum-likelihood estimators of the parameters satisfy the marginal-sum equations. In the present paper we show that this is also true under the more general assumption that the claim numbers of the different cells arise from the decomposition of a collective model for the whole portfolio of risks. In this general setting, the claim numbers of the different cells need not be independent and need not be Poisson distributed.     

Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.     

Uniformly asymptotic behavior for the tail probability of discounted aggregate claims in the time-dependent risk model with upper tail asymptotically independent claims

ABSTRACT

In this paper, we consider the tail behavior of discounted aggregate claims in a dependent risk model with constant interest force, in which the claim sizes are of upper tail asymptotic independence structure, and the claim size and its corresponding inter-claim time satisfy a certain dependence structure described by a conditional tail probability of the claim size given the inter-claim time before the claim occurs. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all times in a finite interval. Moreover, we prove that if the claim size distribution belongs to the consistent variation class, the formula holds uniformly for all times in an infinite interval.     

Precise large deviations for aggregate claims

Abstract

In this article, we consider a non standard renewal risk model, in which the claim sizes form a sequence of independent and identically distributed random variables; the inter-arrival times are negatively associated; and each pair of the claim size and its inter-arrival time follows negative association or arbitrary dependence structure. We establish some precise large-deviation formulas for the aggregate amount of claims in the heavy-tailed case.     

Pólya–Aeppli of Order k Risk Model

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 ? ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.     

BIVARIATE DISTRIBUTION WITH TRUNCATED POISSON MARGINAL DISTRIBUTIONS

ABSTRACT

A bivariate distribution, whose marginal distributions are truncated Poisson distributions, is developed as a product of truncated Poisson distributions and a multiplicative factor. The multiplicative factor takes into account the correlation, either positive or negative, between the two random variables. The distributional properties of this model are studied and the model is fitted to a real life bivariate data.     

A prediction approach for precise marketing based on ARIMA-ARCH Model: A case of China Mobile

The autoregressive integrated moving average (ARIMA) model presents improved performance in forecasting short-term trends because it considers the dependence of time series and the interference of stochastic volatility. Thus, in this study, we establish ARIMA(0, 2, 1) based on the historical data of large-scale online marketing promotions to realize precise marketing of China Mobile's Ling Xi Voice app in the communication market. We eliminate the auto-regression effect of residual series by establishing the ARIMA model combined with the autoregressive conditional heteroskedasticity (ARCH) model denoted as ARIMA(0, 2, 1) ? ARCH(1), the ARIMA model combined with the generalized ARCH (GARCH) model denoted as ARIMA(0, 2, 1) ? GARCH(1, 1), and the ARIMA model combined with the threshold GARCH model denoted as ARIMA(0, 2, 1) ? TGARCH(2, 1). The performance of the aforementioned models is then compared for validation. Considering the characteristics of the communication markets and the attractive statistical properties of ARIMA, we apply ARIMA(0, 2, 1) to forecast the cumulative number of Ling Xi Voice app users for precise marketing that offers reliable agreement for China Mobile to further advertise and study the market demand. Our analysis contributes toward the development of the current knowledge on forecasting the number of app users in the communication market and provides a new idea to increase the market share for communication operators.     

非寿险费率厘定的索赔频率预测模型及其应用

在非寿险分类费率厘定中,泊松回归模型是最常使用的索赔频率预测模型,但实际的索赔频率数据往往存在过离散特征,使泊松回归模型的结果缺乏可靠性.因此,讨论处理过离散问题的各种回归模型,包括负二项回归模型、泊松-逆高斯回归模型、泊松-对数正态回归模型、广义泊松回归模型、双泊松回归模型、混合负二项回归模型、混合二项回归模型、Delaporte回归模型和Sichel回归模型,并对其进行系统比较研究认为:这些模型都可以看做是对泊松回归模型的推广,可以用于处理各种不同过离散程度的索赔频率数据,从而改善费率厘定的效果;同时应用一组实际的汽车保险数据,讨论这些模型的具体应用.     

A Poisson Limit Theorem for Reliability Models Based on Markov Chains

Abstract

In this article, we deal with a class of discrete-time reliability models. The failures are assumed to be generated by an underlying time inhomogeneous Markov chain. The multivariate point process of failures is proved to converge to a Poisson-type process when the failures are rare. As a result, we obtain a Compound Poisson approximation of the cumulative number of failures. A rate of convergence is provided.     

Simplified hierarchical linear models for the evaluation of surrogate endpoints

The linear mixed-effects model (Verbeke and Molenberghs, 2000) has become a standard tool for the analysis of continuous hierarchical data such as, for example, repeated measures or data from meta-analyses. However, in certain situations the model does pose insurmountable computational problems. Precisely this has been the experience of Buyse et al. (2000a) who proposed an estimation- and prediction-based approach for evaluating surrogate endpoints. Their approach requires fitting linear mixed models to data from several clinical trials. In doing so, these authors built on the earlier, single-trial based, work by Prentice (1989), Freedman et al. (1992), and Buyse and Molenberghs (1998). While Buyse et al. (2000a) claim their approach has a number of advantages over the classical single-trial methods, a solution needs to be found for the computational complexity of the corresponding linear mixed model. In this paper, we propose and study a number of possible simplifications. This is done by means of a simulation study and by applying the various strategies to data from three clinical studies: Pharmacological Therapy for Macular Degeneration Study Group (1977), Ovarian Cancer Meta-analysis Project (1991) and Corfu-A Study Group (1995).