A bivariate model of claim frequencies and severities

Bivariate claim data come from a population that consists of insureds who may claim either one, both or none of the two types of benefits covered by a policy. In the present paper, we develop a statistical procedure to fit bivariate distributions of claims in presence of covariates. This allows for a more accurate study of insureds' choice and size in the frequency and severity of the two types of claims. A generalised logistic model is employed to examine the frequency probabilities, whilst the three parameter Burr distribution is suggested to model the underlying severity distributions. The bivariate copula model is exploited in such a way that it allows us to adjust for a range of frequency dependence structures; a method for assessing the adequacy of the fitted severity model is outlined. A health claims dataset illustrates the methods; we describe the use of orthogonal polynomials for characterising the relationship between age and the frequency and severity models.     

On the expected discounted penalty function for a risk model with dependence under a multi-layer dividend strategy

In this article, we consider a dependent risk model in the presence of a multi-laydividend strategy. We construct the dependence structure between the claim size and interclaim time by a Farlie–Gumbel–Morgenstern copula. A piecewise integro-differential equations for the expected discounted penalty function with boundary conditions are established. A renewal equation satisfied by the expected discounted penalty function is obtained via the translation operator. Then, we provide a recursive approach to derive the analytical solution of the expected discounted penalty function. Finally, a numerical example is presented to illustrate the solution procedure.     

Finite time ruin probability and structural density properties in the presence of dependence in insurance risk model

In the present paper, we consider the classical compound Poisson risk model with dependence between claim sizes and claim inter-arrival time. We attempt to analyze the approximation of finite time ruin probability. The finite time ruin probabilities are plotted for fixed threshold value associated to the claim inter-arrival time and also for fixed dependence parameter in Nelsen (2006) copula separately. Additionally, a general form for joint density of the interclaim times and claim sizes is considered. With respect to the classical Gerber-Shiu's (1998) function, first some structural density properties of dependent collective risk model is obtained. Then the ladder height probability density function of claim sizes is computed and the dependency structure investigated for Erlang interclaim time. As the application, some dependent models of the interclaim times and claim sizes are studied.     

Modelling repeated insurance claim frequency data using the generalized linear mixed model

Most of the methods used to estimate claim frequency rates in general insurance have assumed that data are independent. However, it is not uncommon for information stored in the database of an insurance company to contain previous years' claim data from each policyholder. We consider the application of the generalized linear mixed model approach to the analysis of repeated insurance claim frequency data in which a conditionally fixed random effect vector is incorporated explicitly into the linear predictor to model the inherent correlation. A motor insurance data set is used as the basis for simulation to demonstrate the advantages of the method. Ignoring the underlying association for observations within the same policyholder results in an underestimation of the standard error of the parameter estimates and a remarkable reduction in the prediction accuracy. The method provides a viable alternative for incorporating repeated claim experience that enables the revision of rates in general insurance.     

Nonparametric analysis of aggregate loss models

This paper describes a nonparametric approach to make inferences for aggregate loss models in the insurance framework. We assume that an insurance company provides a historical sample of claims given by claim occurrence times and claim sizes. Furthermore, information may be incomplete as claims may be censored and/or truncated. In this context, the main goal of this work consists of fitting a probability model for the total amount that will be paid on all claims during a fixed future time period. In order to solve this prediction problem, we propose a new methodology based on nonparametric estimators for the density functions with censored and truncated data, the use of Monte Carlo simulation methods and bootstrap resampling. The developed methodology is useful to compare alternative pricing strategies in different insurance decision problems. The proposed procedure is illustrated with a real dataset provided by the insurance department of an international commercial company.     

Claim Dependence Induced by Common Effects in Hierarchical Credibility Models

In the usual credibility model, observations are made of a risk or group of risks selected from a population, and claims are assumed to be independent among different risks. However, there are some problems in practical applications and this assumption may be violated in some situations. Some credibility models allow for one source of claim dependence only, that is, across time for an individual insured risk or a group of homogeneous insured risks. Some other credibility models have been developed on a two-level common effects model that allows for two possible sources of dependence, namely, across time for the same individual risk and between risks. In this paper, we argue for the notion of modeling claim dependence on a three-level common effects model that allows for three possible sources of dependence, namely, across portfolios, across individuals and simultaneously across time within individuals. We also obtain the corresponding credibility premiums hierarchically using the projection method. Then we derive the general hierarchical structure or multi-level credibility premiums for the models with h-level of common effects.     

Evaluating the Effectiveness of State-Switching Time Series Models for U.S. Real Output

Two types of state-switching models for U.S. real output have been proposed: models that switch randomly between states and models that switch states deterministically, as in the threshold autoregressive model of Potter. These models have been justified primarily on how well they fit the sample data, yielding statistically significant estimates of the model coefficients. Here we propose a new approach to the evaluation of an estimated nonlinear time series model that provides a complement to existing methods based on in-sample fit or on out-of-sample forecasting. In this new approach, a battery of distinct nonlinearity tests is applied to the sample data, resulting in a set of p-values for rejecting the null hypothesis of a linear generating mechanism. This set of p-values is taken to be a “stylized fact” characterizing the nonlinear serial dependence in the generating mechanism of the time series. The effectiveness of an estimated nonlinear model for this time series is then evaluated in terms of the congruence between this stylized fact and a set of nonlinearity test results obtained from data simulated using the estimated model. In particular, we derive a portmanteau statistic based on this set of nonlinearity test p-values that allows us to test the proposition that a given model adequately captures the nonlinear serial dependence in the sample data. We apply the method to several estimated state-switching models of U.S. real output.     

Multi-type insurance claim processes with high-dimensional covariates

Multi-type insurance claim processes have attracted considerable research interest in the literature. The existing statistical inference for such processes, however, may encounter “curse of dimensionality” due to high-dimensional covariates. In this article, a technique of sufficient dimension reduction is applied to multiple-type insurance claim data, which uses a copula to model the dependence between different types of claim processes, and incorporates a one-dimensional frailty to fit the dependence of claims “within” the same claim process. A two-step procedure is proposed to estimate model parameters. The first step develops nonparametric estimators of the baseline, the basis of the central subspace and its dimension, and the regression function. Then the second step estimates the copula parameter. Simulations are performed to evaluate and confirm the theoretical results.     

Asymptotic Behavior of Random Time Ruin Probability Under Heavy-Tailed Claim Sizes and Dependence Structure

This article deals with the renewal risk model, in which there exists some asymptotic dependence relation between claim sizes and the inter-arrival times, and claim sizes are subexponential. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results.     

Local quantile regression

Quantile regression is a technique to estimate conditional quantile curves. It provides a comprehensive picture of a response contingent on explanatory variables. In a flexible modeling framework, a specific form of the conditional quantile curve is not a priori fixed. This motivates a local parametric rather than a global fixed model fitting approach. A nonparametric smoothing estimator of the conditional quantile curve requires to balance between local curvature and stochastic variability. In this paper, we suggest a local model selection technique that provides an adaptive estimator of the conditional quantile regression curve at each design point. Theoretical results claim that the proposed adaptive procedure performs as good as an oracle which would minimize the local estimation risk for the problem at hand. We illustrate the performance of the procedure by an extensive simulation study and consider a couple of applications: to tail dependence analysis for the Hong Kong stock market and to analysis of the distributions of the risk factors of temperature dynamics.     

A Bayesian method of distinguishing unit root from stationary processes based on panel data models with cross-sectional dependence

In this paper we develop a Bayesian approach to detecting unit roots in autoregressive panel data models. Our method is based on the comparison of stationary autoregressive models with and without individual deterministic trends, to their counterpart models with a unit autoregressive root. This is done under cross-sectional dependence among the error terms of the panel units. Simulation experiments are conducted with the aim to assess the performance of the suggested inferential procedure, as well as to investigate if the Bayesian model comparison approach can distinguish unit root models from stationary autoregressive models under cross-sectional dependence. The approach is applied to real exchange rate series for a panel of the G7 countries and to a panel of US nominal interest rates data.     

Earthquakes occurrences estimation through a parametric semi-Markov approach

The estimation of earthquakes’ occurrences prediction in seismic areas is a challenging problem in seismology and earthquake engineering. Indeed, the prevention and the quantification of possible damage provoked by destructive earthquakes are directly linked to this kind of prevision. In our paper, we adopt a parametric semi-Markov approach. This model assumes that a sequence of earthquakes is seen as a Markov process and besides it permits to take into consideration the more realistic assumption of events’ dependence in space and time. The elapsed time between two consecutive events is modeled as a general Weibull distribution. We determine then the transition probabilities and the so-called crossing states probabilities. We conclude then with a Monte Carlo simulation and the model is validated through a large database containing real data.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

A bivariate regression model with cure fraction

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we introduce a location-scale model for bivariate survival times based on the copula to model the dependence of bivariate survival data with cure fraction. We create the correlation structure between the failure times using the Clayton family of copulas, which is assumed to have any distribution. It turns out that the model becomes very flexible with respect to the choice of the marginal distributions. For the proposed model, we consider inferential procedures based on constrained parameters under maximum likelihood. We derive the appropriate matrices for assessing local influence under different perturbation schemes and present some ways to perform global influence analysis. The relevance of the approach is illustrated using a real data set and a diagnostic analysis is performed to select an appropriate model.     

Measuring density dependence in survival from mark-recapture data

We discuss the analysis of mark-recapture data when the aim is to quantify density dependence between survival rate and abundance. We describe an analysis for a random effects model that includes a linear relationship between abundance and survival using an errors-in-variables regression estimator with analytical adjustment for approximate bias. The analysis is illustrated using data from short-tailed shearwaters banded for 48 consecutive years at Fisher Island, Tasmania, and Hutton's shearwater banded at Kaikoura, New Zealand for nine consecutive years. The Fisher Island data provided no evidence of a density dependence relationship between abundance and survival, and confidence interval widths rule out anything but small density dependent effects. The Hutton's shearwater data were equivocal with the analysis unable to rule out anything but a very strong density dependent relationship between survival and abundance.     

索赔次数的开放式混合泊松分布研究

本文建立了索赔次数的多风险类别混合泊松模型。首先,考虑索赔次数的零膨胀、厚尾性和异质性等特征,建立风险类别待定的开放式混合泊松模型,开放式结构使该模型对实际数据的多样特征和风险类别具有良好的自适应性;其次,定义混合权重参数的iSCAD惩罚函数,实现对权重参数的筛选;最后,借助EM算法求得模型参数,实现对各风险类别下索赔次数的估计。借助iSCAD惩罚函数,给出最优混合数,避免传统混合模型中主观选择的弊端,克服传统混合模型中结构复杂、参数估计没有显式表达式、估计结果不便于解释等问题。基于三组风险特征多样数据的实证分析,本文发现该模型可以显著改进现有模型的拟合效果。     

On the first time of ruin in two-dimensional discrete time risk model with dependent claim occurrences

This article is concerned with a two-dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present numerical results on this model by comparing the corresponding results of the bivariate compound binomial risk model.     

Optimal inference for Simon's two-stage design with over or under enrollment at the second stage

Simon's two-stage designs are widely used in clinical trials to assess the activity of a new treatment. In practice, it is often the case that the second stage sample size is different from the planned one. For this reason, the critical value for the second stage is no longer valid for statistical inference. Existing approaches for making statistical inference are either based on asymptotic methods or not optimal. We propose an approach to maximize the power of a study while maintaining the type I error rate, where the type I error rate and power are calculated exactly from binomial distributions. The critical values of the proposed approach are numerically searched by an intelligent algorithm over the complete parameter space. It is guaranteed that the proposed approach is at least as powerful as the conditional power approach which is a valid but non-optimal approach. The power gain of the proposed approach can be substantial as compared to the conditional power approach. We apply the proposed approach to a real Phase II clinical trial.     

Measuring density dependence in survival from mark-recapture data

We discuss the analysis of mark-recapture data when the aim is to quantify density dependence between survival rate and abundance. We describe an analysis for a random effects model that includes a linear relationship between abundance and survival using an errors-in-variables regression estimator with analytical adjustment for approximate bias. The analysis is illustrated using data from short-tailed shearwaters banded for 48 consecutive years at Fisher Island, Tasmania, and Hutton's shearwater banded at Kaikoura, New Zealand for nine consecutive years. The Fisher Island data provided no evidence of a density dependence relationship between abundance and survival, and confidence interval widths rule out anything but small density dependent effects. The Hutton's shearwater data were equivocal with the analysis unable to rule out anything but a very strong density dependent relationship between survival and abundance.