A new approach to the credibility formula

The usual credibility formula holds whenever, (i) claim size distribution is a member of the exponential family of distributions, (ii) prior distribution conjugates with claim size distribution, and (iii) square error loss has been considered. As long as, one of these conditions is violent, the usual credibility formula no longer holds. This article, using the mean square error minimization technique, develops a simple and practical approach to the credibility theory. Namely, we approximate the Bayes estimator with respect to a general loss function and general prior distribution by a convex combination of the observation mean and mean of prior, say, approximate credibility formula. Adjustment of the approximate credibility for several situations and its form for several important losses are given.     

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to $C^{1,\varsigma }$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard’s regularity theorem in the sense that the special time-independent case reduces to Allard’s theorem.     

Second Order Unconditionally Stable and Convergent Linearized Scheme for a Fluid-Fluid Interaction Model

In this paper, a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model, which consists of two Navier-Stokes equations coupled by a linear interface condition. The proposed fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the second-order backward differentiation formula for temporal discretization, the second-order Gear's extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms. Moreover, the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.     

相对损失函数下的倍度保费

传统的倍度保费公式利用均方损失函数估计特定保人的风险. 然而, 索取保费与真实保费之间的比例比它们差的绝对值更适合于衡量保费的公平性. 基于这一点, 我们提出了两种计算保费的损失函数: 均方相对损失函数和熵相对损失函数, 并且给出了倍度因子的估计公式及它们的性质.     

Second Order Cones for Maximal Monotone Operators via Representative Functions

It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T. The second author’s research was funded by NSERC and the Canada Research Chair programme, and the first author’s by ARC grant number DP0664423. This study was commenced between August and December 2005 while the first author was visiting Dalhousie University.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping

Some new oscillation criteria are established for the nonlinear damped differential equation (r(t)y′)′+p(t)y′+q(t)f(y)=0 that are different from most known ones in the sense that they are based on a class of new functions Φ(t,s,r) defined in the sequel. Our results are sharper than some previous results which can be seen by the examples at the end of this paper.     

二阶中立型微分方程的区间振动准则

利用平均函数技巧 ,对二阶中立型微分方程 [a(t) (x(t) p(t)x(t-τ) )′]′ q(t)f[x(t) ,x(t-σ) ]g[x′(t) ]=0建立了一些区间振动准则 ,这些振动准则不同于已知依赖于整个区间 [t0 ,∞ )的性质的结果 ,而是仅依赖于 [t0 ,∞ )上的子区间列的性质     

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Besides the claims data in the past, certain assumptions about the distribution of claimsare required to derive the credibility premium in the classical theory. In the paper, the credibility premium can be calculated via the maximum entropy method if we know nothing about the distribution of claims. Furthermore, two corollaries are obtained under certain assumptions, that is, new claims have more weight than the old ones and the classical credibility formula is a special case of the credibility premium derived in the present paper. Finally, the simulation study is presented to illustrate that the credibility premium in the present paper is better than other models if the mean square error is taken as the evaluation criterion.     

二阶半线性常微分方程的区间振动准则

通过利用平均函数及 Ｈａｒｄｙ ，Ｌｉｔｔｌｅｗｏｏｄ和 Ｐｏｌｙａ不等式，对二阶半线性微分方程建立了一些新的区间振动准则，这些准则不同于已知的依赖于整个区间[ｔ０，∞）的性质的结果，而是仅依赖于区间[ｔ０，∞）的子区间列的性质．所得结果推广和改进了 Ｋａｍｅｎｅｖ，Ｋｏｎｇ，Ｌｉ和 Ｙｅｈ及 Ｐｈｉｌｏｓ的振动准则，同时也可应用于以前所不能处理的某些情形，如特别，还给出了几个例子以说明本文所得结果优越性．     

帕累托索赔分布中风险参数的经验贝叶斯估计

本文建立了贝叶斯模型,讨论了帕累托索赔额分布中参数的估计问题,得到了风险参数的极大似然估计、贝叶斯估计和信度估计,并证明了这些估计的强相合性.在均方误差的意义下比较了这些估计的好坏,并通过数值模拟对均方误差进行了验证,结果表明,贝叶斯估计比其他估计具有较小的均方误差.最后,给出了结构参数的估计并证明了经验贝叶斯估计和经验贝叶斯信度估计的渐近最优性.     

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.     

A generalization of the credibility theory obtained by using the weighted balanced loss function

In this paper an alternative to the usual credibility premium that arises for weighted balanced loss function is considered. This is a generalized loss function which includes as a particular case the weighted quadratic loss function traditionally used in actuarial science. From this function credibility premiums under appropriate likelihood and priors can be derived. By using weighted balanced loss function we obtain, first, generalized credibility premiums that contain as particular cases other credibility premiums in the literature and second, a generalization of the well-known distribution free approach in [Bühlmann, H., 1967. Experience rating and credibility. Astin Bull. 4 (3), 199-207].     

Tolerance intervals in risk theory

Up to now, the main aim of credibility theory has been to provide statistical models which allow for estimating (net) risk premiums appropriately. In the present note, a simple credibility model based on the percentile principle is introduced. It turns out that there are close connections between the resulting credibility premiums and statistical tolerance limits.     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

An unbayesed approach to credibility

It is shown how a classical credibility formula can be obtained by alternative methods, the Straub approach and the direct approach, none of which relies on the Bayes technique.     

基于可信性分布的投资组合问题的一种混合智能方法

研究了模糊环境下基于效用函数的有效资产投资组合的收益率模型,模型建立在可信性分布的基础上,而不是概率分布或可能性分布基础上.给出模糊环境下基于可信性分布的n种资产的最优投资组合问题的混合智能算法以寻找某种效用函数意义下的最优组合.并以实例仿真说明该方法的有效性.     

Optimal robust insurance with a finite uncertainty set

Decision-makers who usually face model/parameter risk may prefer to act prudently by identifying optimal contracts that are robust to such sources of uncertainty. In this paper, we tackle this issue under a finite uncertainty set that contains a number of probability models that are candidates for the “true”, but unknown model. Various robust optimisation models are proposed, some of which are already known in the literature, and we show that all of them can be efficiently solved via Second Order Conic Programming (SOCP). Numerical experiments are run for various risk preference choices and it is found that for relatively large sample size, the modeler should focus on finding the best possible fit for the unknown probability model in order to achieve the most robust decision. If only small samples are available, then the modeler should consider two robust optimisation models, namely the Weighted Average Model or Weighted Worst-case Model, rather than focusing on statistical tools aiming to estimate the probability model. Amongst those two, the better choice of the robust optimisation model depends on how much interest the modeler puts on the tail risk when defining its objective function. These findings suggest that one should be very careful when robust optimal decisions are sought in the sense that the modeler should first understand the features of its objective function and the size of the available data, and then to decide whether robust optimisation or statistical inferences is the best practical approach.     

Credibility evaluation for the exponential dispersion family

It has long been established that under regularity conditions, the linear credibility formula with an appropriate credibility factor produces exact fair premium for claims or losses whose distribution is a member of the natural exponential family. Recently, this result has been extended to a richer family of distribution, the exponential dispersion family which comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter σ²=1/λ which is equal to 1 in the case of the natural exponential family. In this paper neither λ is regarded known, nor a fully specified prior distribution for λ is assumed. Instead, by establishing a link between the m.s.e. of the linear credibility and Fisher information we derive optimal credibility for the case where only the mean and variance of λ are specified.