Addendum to ‘The multi-year non-life insurance risk in the additive reserving model’ [Insurance Math. Econom. 52(3) (2013) 590–598]: Quantification of multi-year non-life insurance risk in chain ladder reserving models

This is the first study to derive closed-form analytical expressions for multi-year non-life insurance risk in the chain ladder model. Extending on previous research on the additive reserving model, we define multi-year risk via prediction errors of multi-year claims development results including both observed and future accident years. A resampling argument and a first-order Taylor approximation address the quantification of estimation errors and multiplicative dependencies in the chain ladder framework, respectively. From our generalized multi-year approach, we deduce estimators for reserve and premium risks in multi-year view and their implicit correlation. We reproduce well-known results from literature for the special cases of one-year and ultimo view. Further, we comment on how to obtain estimators for generalized versions of the chain ladder method. A case study demonstrates the applicability of our analytical formulae.     

Robust loss reserving in a log-linear model

It is well known that the presence of outlier events can overestimate or underestimate the overall reserve when using the chain-ladder method. The lack of robustness of loss reserving estimators leads to the development of this paper. The appearance of outlier events (including large claims—catastrophic events) can offset the result of the ordinary chain ladder technique and perturb the reserving estimation. Our proposal is to apply robust statistical procedures to the loss reserving estimation, which are insensitive to the occurrence of outlier events in the data. This paper considers robust log-linear and ANOVA models to the analysis of loss reserving by using different type of robust estimators, such as LAD-estimators, M-estimators, LMS-estimators, LTS-estimators, MM-estimators (with initial S-estimate) and Adaptive-estimators. Comparisons of these estimators are also presented, with application of a well known data set.     

A general multivariate chain ladder model

A general multivariate stochastic reserving model is formulated, which not only specifies contemporaneous correlations, but also allows structural connections among triangles. Its structure extends the existing multivariate chain ladder models in a natural way, and this extension proves to be advantageous in improving model adequacy and increasing model flexibility. It is general in the sense that it includes various models in the chain ladder framework as special cases. At the heart of this model is the seemingly unrelated regression technique, which is utilized to estimate parameters that reflect contemporaneous correlations. The use of this technique is essential to construct flexible models, and related statistical theories are applied to study properties of existing estimators. A numerical example is utilized to show the advantage of the proposed model in studying multiple triangles that are related both structurally and contemporaneously.     

The collective reserving model

This paper sets out a model for analysing claims development data, which we call the collective reserving model (CRM). The model is defined on the individual claim level and it produces separate IBNR and RBNS reserve estimators at the collective level without using any approximations. The CRM is based on ideas from a paper by Verrall, Nielsen and Jessen (VNJ) from 2010 in which a model is proposed that relies on a claim giving rise to a single payment. This is generalised by the CRM to the case of multiple payments per claim. All predictors of outstanding claims payments for the VNJ model are shown to hold for this new model. Moreover, the quasi-Poisson GLM estimation framework will be applicable as well, but without using an approximation. Furthermore, analytical expressions for the variance of the total outstanding claims payments are given, with a subdivision on IBNR and RBNS claims. To quantify the effect of allowing only one payment per claim, the model is related and compared to the VNJ model, in particular by looking at variance inequalities. The double chain ladder (DCL) method is discussed as an estimation method for this new model and it is shown that both the GLM- and DCL-based estimators are consistent in terms of an exposure measure. Lastly, both of these methods are shown to asymptotically reproduce the regular chain ladder reserve estimator when restricting predictions to the lower right triangle without the tail, motivating the chain ladder technique as a large-exposure approximation of this model.     

基于伽玛与对数正态分布假设下的广义线性模型的比较和应用

本文在假设每次损失金额的变异系数相同,且它们服从伽玛分布或对数正态分布的条件下,讨论了加法模型和乘法模型的参数估计和拟合优度检验,并应用一组实际损失数据对上述模型进行了实证比较。结果表明,对于一组特定的损失数据,对数正态分布假设下的广义线性模型可能优于伽玛分布假设下的广义线性模型。     

Minimax estimator of regression coefficient in normal distribution under balanced loss function

This article investigates linear minimax estimators of regression coefficient in a linear model with an assumption that the underlying distribution is a normal one with a nonnegative definite covariance matrix under a balanced loss function. Some linear minimax estimators of regression coefficient in the class of all estimators are obtained. The result shows that the linear minimax estimators are unique under some conditions.     

All admissible linear estimators of a regression coefficient under a balanced loss function

Admissibility of linear estimators of a regression coefficient in linear models with and without the assumption that the underlying distribution is normal is discussed under a balanced loss function. In the non-normal case, a necessary and sufficient condition is given for linear estimators to be admissible in the space of homogeneous linear estimators. In the normal case, a sufficient condition is provided for restricted linear estimators to be admissible in the space of all estimators having finite risks under the balanced loss function. Furthermore, the sufficient condition is proved to be necessary in the normal case if additional conditions are assumed.     

Preliminary test and Stein estimations in simultaneous linear equations

In modeling of an economic system, there may occur some stochastic constraints, that can cause some changes in the estimators and their respective behaviors. In this approach we formulate the simultaneous equation models into the problem of estimating the regression parameters of a multiple regression model, under elliptical errors. We define five different sorts of estimators for the vector-parameter. Their exact risk expressions are also derived under the balanced loss function. Comparisons are then made to clarify the performance of the proposed estimators. It is shown that the shrinkage factor of the Stein estimator is robust with respect to departures from normality assumption.     

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.     

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

A copula based Bayesian approach for paid–incurred claims models for non-life insurance reserving

Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of hierarchical Bayesian paid–incurred claims models, combining the claims reserving models of Hertig (1985) and Gogol (1993). In the process we extend the independent log-normal model of Merz and Wüthrich (2010) by incorporating different dependence structures using a Data-Augmented mixture Copula paid–incurred claims model.In this way the paper makes two main contributions: firstly we develop an extended class of model structures for the paid–incurred chain ladder models where we develop precisely the Bayesian formulation of such models; secondly we explain how to develop advanced Markov chain Monte Carlo sampling algorithms to make inference under these copula dependence PIC models accurately and efficiently, making such models accessible to practitioners to explore their suitability in practice. In this regard the focus of the paper should be considered in two parts, firstly development of Bayesian PIC models for general dependence structures with specialised properties relating to conjugacy and consistency of tail dependence across the development years and accident years and between Payment and incurred loss data are developed. The second main contribution is the development of techniques that allow general audiences to efficiently work with such Bayesian models to make inference. The focus of the paper is not so much to illustrate that the PIC paper is a good class of models for a particular data set, the suitability of such PIC type models is discussed in Merz and Wüthrich (2010) and Happ and Wüthrich (2013). Instead we develop generalised model classes for the PIC family of Bayesian models and in addition provide advanced Monte Carlo methods for inference that practitioners may utilise with confidence in their efficiency and validity.     

Improved estimation in multiple linear regression models with measurement error and general constraint

In this paper, we define two restricted estimators for the regression parameters in a multiple linear regression model with measurement errors when prior information for the parameters is available. We then construct two sets of improved estimators which include the preliminary test estimator, the Stein-type estimator and the positive rule Stein type estimator for both slope and intercept, and examine their statistical properties such as the asymptotic distributional quadratic biases and the asymptotic distributional quadratic risks. We remove the distribution assumption on the error term, which was generally imposed in the literature, but provide a more general investigation of comparison of the quadratic risks for these estimators. Simulation studies illustrate the finite-sample performance of the proposed estimators, which are then used to analyze a dataset from the Nurses Health Study.     

Assessing Convergence and Mixing of MCMC Implementations via Stratification

Some posterior distributions lead to Markov chain Monte Carlo (MCMC) chains that are naturally viewed as collections of subchains. Examples include mixture models, regime-switching models, and hidden Markov models. We obtain MCMC-based estimators of posterior expectations by combining different subgroup (subchain) estimators using stratification and poststratification methods. Variance estimates of the limiting distributions of such estimators are developed. Based on these variance estimates, we propose a test statistic to aid in the assessment of convergence and mixing of chains. We compare our diagnostic with other commonly used methods. The approach is illustrated in two examples: a latent variable model for arsenic concentration in public water systems in Arizona and a Bayesian hierarchical model for Pacific sea surface temperatures. Supplementary materials, which include MATLAB codes for the proposed method, are available online.     

Sample rotation theory with missing data

This paper studies how the sample rotation method is applied to the case where item non-response occurs in surveys. The two cases where the response to the first occasion is complete or incomplete are considered. Using ratio imputation method, the estimators of the current population mean are proposed, which are valid under uniform response regardless of the model and under the ratio model regardless of the response mechanism. Under uniform response, the variances of the proposed estimators are derived. Interestingly, although their expressions are similar, the estimator for the case of incomplete response on the first occasion can have smaller variance than the one for the case of complete response on the first occasion under uniform response. The linearized jackknife variance estimators are also given. These variance estimators prove to be approximately design-unbiased under uniform response. It should be noted that similar property on variance estimators has not been discussed in literature.     

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution

Starting from the question: What is the accident risk of an insured individual?, we consider that the customer has contracted policies in different insurance lines: motor and home. Three models based on the multivariate Sarmanov distribution are analyzed. Driven by a real data set that takes into account three types of accident risks, two for motor and one for home, three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals are considered and fitted to the data. To estimate the parameters of these three models, we discuss a method for approaching the maximum likelihood (ML) estimators. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM and with elliptical copula based models.     

Existence conditions for the uniformly minimum risk unbiased estimators in a class of linear models

This paper studies the existence of the uniformly minimum risk unbiased (UMRU) estimators of parameters in a class of linear models with an error vector having multivariate normal distribution or t-distribution, which include the growth curve model, the extended growth curve model, the seemingly unrelated regression equations model, the variance components model, and so on. The necessary and sufficient existence conditions are established for UMRU estimators of the estimable linear functions of regression coefficients under convex losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model with normality assumption, the conclusions given in the literature can be derived by applying the general results in this paper. For the variance components model, the necessary and sufficient existence conditions are reduced as terse forms.     

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??The Bayes estimators of variance components are derived under weighted square loss function for the balanced one-way classification random effects model with the assumption that variance component has the conjugate prior distribution. The superiorities of the Bayes estimators for variance components to traditional ANOVA estimators are studied in terms of the mean square error (MSE) criterion. Finally, a remark for main results is given.     

Semiparametric model for prediction of individual claim loss reserving

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Conventional methods, such as ladder or separation methods based on aggregated or grouped claims of the so-called “run-off triangle”, have been illustrated to have some drawbacks. Recently, individual claim loss models have attracted a great deal of interest in actuarial literature, which can overcome the shortcomings of aggregated claim loss models. In this paper, we propose an alternative individual claim loss model, which has a semiparametric structure and can be used to fit flexibly the claim loss reserving. Local likelihood is employed to estimate the parametric and nonparametric components of the model, and their asymptotic properties are discussed. Then the prediction of the IBNR claim loss reserving is investigated. A simulation study is carried out to evaluate the performance of the proposed methods.     

Bayesian estimation and inference for log-ACD models

This paper adapts Bayesian Markov chain Monte Carlo methods for application to some auto-regressive conditional duration models. Subsequently, the properties of these estimators are examined and assessed across a range of possible conditional error distributions and dynamic specifications, including under error mis-specification. A novel model error distribution, employing a truncated skewed Student-t distribution is proposed and the Bayesian estimator assessed for it. The results of an extensive simulation study reveal that favourable estimation properties are achieved under a range of possible error distributions, but that the generalised gamma distribution assumption is most robust and best preserves these properties, including when it is incorrectly specified. The results indicate that the powerful numerical methods underlying the Bayesian estimator allow more efficiency than the (quasi-) maximum likelihood estimator for the cases considered.     

On the optimal design of the randomized unbiased Monte Carlo estimators

We consider a class of unbiased Monte Carlo estimators and develop an efficient algorithm to produce the distribution of the optimal random truncation level. We establish the convergence and optimality results of the associated algorithm and also derive its exact complexity. We find this algorithm has a much lower complexity as compared to the existing one in the literature. The proposed algorithm is also applicable to optimization problems arising in supply chain management, such as economic reorder interval problem.