On some multivariate Sarmanov mixed Erlang reinsurance risks: Aggregation and capital allocation

Following some recent works on risk aggregation and capital allocation for mixed Erlang risks joined by Sarmanov’s multivariate distribution, in this paper we present some closed-form formulas for the same topic by considering, however, a different kernel function for Sarmanov’s distribution, not previously studied in this context. The risk aggregation and capital allocation formulas are derived and numerically illustrated in the general framework of stop-loss reinsurance, and then in the particular case with no stop-loss reinsurance. A discussion of the dependency structure of the considered distribution, based on Pearson’s correlation coefficient, is also presented for different kernel functions and illustrated in the bivariate case.     

Capture-Recapture模型与车险事故总体预测

车险事故总体预测问题一直是车辆保险公司研究的重点内容之一,目前最为常用的方法是与泊松分布相关的模型.基于车辆保险中索赔数据的结构特征,构建了Capture-Recapture模型,并使用一组车辆保险数据,利用Capture-Recapture及常用的零膨胀泊松等模型分别建模分析,得出了一些新的结论,即Capture-Recapture模型拟合效果整体较优,从而为车辆保险公司更好预测事故总体提供一定的理论依据.     

A note on the Sarmanov bivariate distributions

We investigate the maximum correlation for Sarmanov bivariate distributions with fixed marginals and strengthen the existing results in the literature. The improvement in the maximum correlation is significant. A characterization of the Sarmanov distribution via chi-square divergence is also given. This extends Nelsen [13] result about the Farlie-Gumbel-Morgenstern (FGM) distribution.     

Gregory Solid Construction for Polyhedral Volume Parameterization by Sparse Optimization

In isogeometric analysis, it is frequently required to handle the geometric models enclosed by four-sided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second,the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers(ADMM)is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and effciency of the developed method.     

Sarmanov family of multivariate distributions for bivariate dynamic claim counts model

To predict future claims, it is well-known that the most recent claims are more predictive than older ones. However, classic panel data models for claim counts, such as the multivariate negative binomial distribution, do not put any time weight on past claims. More complex models can be used to consider this property, but often need numerical procedures to estimate parameters. When we want to add a dependence between different claim count types, the task would be even more difficult to handle. In this paper, we propose a bivariate dynamic model for claim counts, where past claims experience of a given claim type is used to better predict the other type of claims. This new bivariate dynamic distribution for claim counts is based on random effects that come from the Sarmanov family of multivariate distributions. To obtain a proper dynamic distribution based on this kind of bivariate priors, an approximation of the posterior distribution of the random effects is proposed. The resulting model can be seen as an extension of the dynamic heterogeneity model described in Bolancé et al. (2007). We apply this model to two samples of data from a major Canadian insurance company, where we show that the proposed model is one of the best models to adjust the data. We also show that the proposed model allows more flexibility in computing predictive premiums because closed-form expressions can be easily derived for the predictive distribution, the moments and the predictive moments.     

Capital Allocation for Sarmanov’s Class of Distributions

This paper is a follow-up of the study realized by Vernic (2014) on the aggregation of dependent random variables joined by Sarmanov’s multivariate distribution, with accent on the particular case of exponentially distributed marginals. More precisely, in this paper we present capital allocation formulas for a portfolio of risks following the just mentioned Sarmanov’s distribution. The overall capital and its allocation to the risk sources are evaluated using the TVaR rule. The resulting formulas are illustrated in some particular cases.     

On the spectral problem for trivariate functions

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun3; a part of Chebfun for numerical computing with 3D functions. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate.     

On the Distribution of a Sum of Sarmanov Distributed Random Variables

Built from given marginals with a flexible dependency structure, Sarmanov’s family of multivariate distributions became of interest in various fields. In this paper, we present some formulas for the density of the sum of several random variables joined by Sarmanov’s distribution, with accent on the particular case of exponentially distributed marginals. Such results are useful in solving, e.g., financial and actuarial problems.     

偏t分布假设下的空间效应模型及其应用

在非寿险索赔强度预测中,目前使用最为广泛的是广义线性模型。索赔强度的广义线性模型假设因变量服从伽马分布或逆高斯分布,且在预测项中仅能考虑协变量的线性效应。这些限制性条件都有可能影响索赔强度预测结果的准确性。本文对索赔强度的广义线性模型进行了推广:用偏T分布代替常用的伽马分布和逆高斯分布;在预测项中引入惩罚样条函数来描述连续型协变量的非线性效应;考虑索赔强度在不同地区的差异性和相邻地区的相依性。最后基于一组实际的车损险数据进行了实证研究,结果表明,本文的推广模型可以明显提高索赔强度预测模型的拟合优度。     

Compound negative binomial shared frailty models for bivariate survival data

In this paper, we introduce a new shared frailty model called the compound negative binomial shared frailty model with three different baseline distributions namely, Weibull, generalized exponential and exponential power distribution. To estimate the parameters involved in these models we adopt Markov Chain Monte Carlo (MCMC) approach. Also we apply these three models to a real life bivariate survival data set of McGrilchrist and Aisbett (1991) related to kidney infection and suggest a better model for the data.     

The loss given default of a low-default portfolio with weak contagion

In this paper we study the loss given default (LGD) of a low default portfolio (LDP), assuming that there is weak credit contagion among the obligors. We characterize the credit contagion by a Sarmanov dependence structure of the risk factors that drive the obligors’ default, where the risk factors are assumed to be heavy tailed. From a new perspective of asymptotic analysis, we derive a limiting distribution for the LGD. As a consequence, an approximation for the entire distribution, in contrast to just the tail behavior, of the LGD is obtained. We show numerical examples to demonstrate the limiting distribution. We also discuss possible applications of the limiting distribution to the calculation of moments and the Value at Risk (VaR) of the LGD.     

On tests of independence in a trivariate exponential distribution

In this paper we consider the problem of testing a null hypothesis H₀ against an alternative hypothesis H_a, related to a trivariate model. H₀ is transformed into a hypothesis concerning the equality of scale parameters of three exponential distributions, following Bhattacharyya and Johnson. Two conditional tests for the transformed hypothesis are considered, and simple expressions for the power functions of our tests are obtained.     

Trivariate near-best blending spline quasi-interpolation operators

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.     

Rational General Solutions of Trivariate Rational Differential Systems

We generalize the method of Ngô and Winkler (J Symbolic Comput 46:1173–1186, 2011) for finding rational general solutions of a plane rational differential system to the case of a trivariate rational differential system. We give necessary and sufficient conditions for the trivariate rational differential system to have a rational solution based on proper reparametrization of invariant algebraic space curves. In fact, the problem for computing a rational solution of the trivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation. We prove that the linear rational solvability of this autonomous differential equation does not depend on the choice of proper parametrizations of invariant algebraic space curves. In addition, two different rational solutions corresponding to the same invariant algebraic space curve are related by a shifting of the variable. We also present a criterion for a rational solution to be a rational general solution.     

A CS-AdaBoost-BP model for product quality inspection

Quality inspection is essential in preventing defective products from entering the market. Due to the typically low percentage of defective products, it is generally challenging to detect them using algorithms that aim for the overall classification accuracy. To help solve this problem, we propose an ensemble learning classification model, where we employ adaptive boosting (AdaBoost) to cascade multiple backpropagation (BP) neural networks. Furthermore, cost-sensitive (CS) learning is introduced to adjust the loss function of the basic classifier of the BP neural network. For clarity, this model is called a CS-AdaBoost-BP model. To empirically verify its effectiveness, we use data from home appliance production lines from Bosch. We carry out tenfold cross-validation to evaluate and compare the performance between the CS-AdaBoost-BP model and three existing models: BP neural network, BP neural network based on sampling, and AdaBoost-BP. The results show that our proposed model not only performs better than the other models but also significantly improves the ability to identify defective products. Furthermore, based on the mean value of the Youden index, our proposed model has the highest stability.

     

An adaptive trivariate dimension-reduction method for statistical moments assessment and reliability analysis

An adaptive trivariate dimension-reduction method is proposed for statistical moments evaluation and reliability analysis in this paper. First, the raw moments of the performance function can be estimated by means of the trivariate dimension-reduction method, where the trivariate, bivariate and univariate Gaussian-weighted integrals are involved. Since the trivariate and bivariate integrals control the efficiency and accuracy, delineating the existence of bivariate and trivariate cross terms is performed, which could significantly reduce the numbers of trivariate and bivariate integrals to be evaluated. When the cross terms exist, the trivariate and bivariate integrals are numerically evaluated directly by the high-order unscented transformation, where the involved free parameters are provided. When the cross terms don’t exist, the trivariate and bivariate integrals can be further decomposed to be the lower-dimensional integrals, where the high-order unscented transformation is again adopted for numerical integrations. In that regard, the first-four central moments can be computed accordingly and the performance function’s probability density function can be reconstructed by fitting the shifted generalized lognormal distribution model based on the first-four central moments. Then, the failure probability can be computed by a one-dimensional integral over the performance function’s probability density function in the failure domain. Three numerical examples, including both the explicit and implicit performance functions, are investigated, to demonstrate the efficacy of the proposed method for both the statistical moments assessment and reliability analysis.     

Quantifying the risk using copulae with nonparametric marginals

We show that copulae and kernel estimation can be mixed to estimate the risk of an economic loss. We analyze the properties of the Sarmanov copula. We find that the maximum pseudo-likelihood estimation of the dependence parameter associated with the copula with double transformed kernel estimation to estimate marginal cumulative distribution functions is a useful method for approximating the risk of extreme dependent losses when we have large data sets. We use a bivariate sample of losses from a real database of auto insurance claims.     

生产安全事故宏观发展态势的计量经济学研究

为了研究工矿商贸就业人员10万人生产安全事故死亡率时间序列变化特征,基于我国行业生产安全事故死亡人数及第二、三产业就业人员数量等2方面年度统计数据,通过研究事故死亡率时间序列的自回归移动平均过程,论文构建了事故死亡率时间序列的分阶段自回归移动平均模型.研究表明:工矿商贸行业10万人事故死亡率变化趋势具有明显的分阶段波动特征,事故死亡率序列均为趋势平稳过程;序列当期观测值与滞后1期观测值具有显著的自相关性;各阶段事故率自回归移动平均模型结构不尽相同;特征描述模型为正确把握我国安全生产状况及趋势提供理论依据.     

Stochastic ordering of medians in exchangeable trivariate normal vectors

In this paper, we establish the stochastic ordering of median from an exchangeable trivariate normal vector based on the strength of the correlation coefficient. Specifically, by considering two exchangeable trivariate normal vectors with different correlation coefficients, we show that the absolute value of the median in the vector with smaller correlation coefficient is stochastically smaller than the absolute value of the median in the vector with larger correlation coefficient. We prove this result by utilizing skew-normal distributions.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.