Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues.In this article,we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences.A specific sub-model form of our suggested family,named as a new extended heavy-tailed Weibull distribution is examined in detail.Some basic characterizations,including quantile function and raw moments have been derived.The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method.To judge the performance of the maximum likelihood estimators,a simulation analysis is performed in detail.Furthermore,some important actuarial measures such as value at risk and tail value at risk are also computed.A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed.The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set.The practical application shows that the proposed model is more flexible and efficient than the other six competing models including(i)the two-parameter models Weibull,Lomax and Burr-XII distributions(ii)the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions,and(iii)a well-known four-parameter Kumaraswamy Weibull distribution.

Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions

A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. The model considers radially symmetric Gaussian, exponential and power-law functions inn= 1, 2, 3 dimensions. Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. For two-dimensional Gaussians and one-dimensional exponential functions, exact power-laws with exponent –1 are obtained. In other cases, densities with an approximate power-law behaviour close to the origin arise. These densities are analyzed using Padé approximants in order to show the approximate power-law behaviour. If the sampled function itself follows a power-law with exponent –α, random sampling leads to densities that also follow an exact power-law, with exponent -n/a – 1. The presented mechanism shows that power-laws can arise in generic situations different from previously considered specialized systems such as multi-particle systems close to phase transitions, dynamical systems at bifurcation points or systems displaying self-organized criticality. Thus, the presented mechanism may serve as an alternative hypothesis in system identification problems.

Large Deviations for Random Sums on Some Kind of Heavy-tailed Classes in Risk Models

This paper is a further investigation into the large deviations for random sums of heavy-tailed,we extended and improved some results in ref. [1] and [2]. These results can applied to some questions in Insurance and Finance.

RUIN PROBABILITY IN A SEMI-MARKOV RISK MODEL WITH CONSTANT INTEREST FORCE AND HEAVY-TAILED CLAIMS

In the present paper, we consider a kind of semi-Markov risk model （SMRM） with constant interest force and heavy-tailed claims~ in which the claim rates and sizes are conditionally independent, both fluctuating according to the state of the risk business. First, we derive a matrix integro-differential equation satisfied by the survival probabilities. Second, we analyze the asymptotic behaviors of ruin probabilities in a two-state SMRM with special claim amounts. It is shown that the asymptotic behaviors of ruin probabilities depend only on the state 2 with heavy-tailed claim amounts, not on the state 1 with exponential claim sizes.

HAZARD FUNCTION AND CHARACTERIZATIONS ON DISTRIBUTION TAILS OF NONNEGATIVE RANDOM VARIABLES

Some equivalent conditions on the classes of lighted-tailed and heavily heavy-tailed and lightly heavy-tailed d.f.s are introduced.The limit behavior of xα(x) and e λx(x) are discussed.Some properties of the subclass DKc and subclass DK 1 are obtained.

Large Deviations for Sums of Heavy-tailed Random Variables

This paper is a further investigation of large deviations for sums of random variables Sn=i=1∑n Xi and S（t）=i=1∑N（t）Xi,（t≥0）, where {X_n,n≥1） are independent identically distribution and non-negative random variables, and {N（t）,t≥0} is a counting process of non-negative integer-valued random variables, independent of {X_n,n≥1}. In this paper, under the suppose F∈G, which is a bigger heavy-tailed class than C, proved large deviation results for sums of random variables.

Gaussian-Student's t mixture distribution PHD robust filtering algorithm based on variational Bayesian inference

Aiming at the problem of filtering precision degradation caused by the random outliers of process noise and measurement noise in multi-target tracking(MTT) system,a new Gaussian-Student’s t mixture distribution probability hypothesis density(PHD) robust filtering algorithm based on variational Bayesian inference(GST-vbPHD) is proposed.Firstly,since it can accurately describe the heavy-tailed characteristics of noise with outliers,Gaussian-Student’s t mixture distribution is employed to model process noise and measurement noise respectively.Then Bernoulli random variable is introduced to correct the likelihood distribution of the mixture probability,leading hierarchical Gaussian distribution constructed by the Gaussian-Student’s t mixture distribution suitable to model non-stationary noise.Finally,the approximate solutions including target weights,measurement noise covariance and state estimation error covariance are obtained according to variational Bayesian inference approach.The simulation results show that,in the heavy-tailed noise environment,the proposed algorithm leads to strong improvements over the traditional PHD filter and the Student’s t distribution PHD filter.

Characterizations on Heavy-tailed Distributions by Means of Hazard Rate

Abstract Let F(x) be a distribution function supported on [0,X), with an equilibrium distribution function Fe(x). In this paper we shall study the function $r_e(x)( - {\rm ln}{\overline F}_e ( x ))\prime = {\overline F}( x )/\int_x^\infty {\overline F}( u )du $, which is called the equilibrium hazard rate of F. By the limiting behavior of re(x) we give a criterion to identify F to be heavy-tailed or light-tailed. Two broad classes of heavy-tailed distributions are also introduced and studied.

Asymptotic behavior for sums of non-identically distributed random variables

For any given positive integer m, let X_i, 1 ≤ i ≤ m be m independent random variables with distributions F_i, 1 ≤ i ≤ m. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio ■equals 1 as x →∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.

An independent but not identically distributed bit error model for heavy-tailed wireless channels

The error patterns of a wireless channel can be represented by a binary sequence of ones(burst) and zeros(run),which is referred to as a trace.Recent surveys have shown that the run length distribution of a wireless channel is an intrinsically heavy-tailed distribution.Analytical models to characterize such features have to deal with the trade-off between complexity and accuracy.In this paper,we use an independent but not identically distributed(inid) stochastic process to characterize such channel behavior and show how to parameterize the inid bit error model on the basis of a trace.The proposed model has merely two parameters both having intuitive meanings and can be easily figured out from a trace.Compared with chaotic maps,the inid bit error model is simple for practical use but can still be deprived from heavy-tailed distribution in theory.Simulation results demonstrate that the inid model can match the trace,but with fewer parameters.We then propose an improvement on the inid model to capture the 'bursty' nature of channel errors,described by burst length distribution.Our theoretical analysis is supported by an experimental evaluation.

微动工作台中弹性导轨的设计计算

介绍了微动工作台中弹性板簧导轨的设计方法、特性误差的计算 ,重点叙述了载荷反对称布置的弹性导轨使微动台产生独立转动自由度的方法 .与以柔性铰链为主体的平面三自由度微动工作台相比 ,弹性板簧导轨式的微动工作台力学模型简单 ,无原理误差 ,从而提高了工作台的位移精度 ,避免工作台在各运动方向之间的耦合 .在板簧上粘贴应变片还可以测出微动台的位移 ,简化了测量系统 .

中国期货合约动态保证金水平的设定研究

针对我国期货收益厚尾分布的非对称性,利用基于极值理论的修正Hill估计与学生分布相结合的VaR-x方法,对我国期货市场的交易保证金水平进行了动态设定.实证结果显示:修正Hill估计与学生分布相结合的动态GARCH-VaR-x模型比EWMA-VaR-x模型更能提供精确的交易保证金预测;与静态调整的保证金收取方式相比,动态交易保证金能更好地捕捉极端风险,可确保在较高风险覆盖率情况下能够收取更少的保证金.

An asymptotic relationship for ruin probabilities under heavy-tailed claims

The famous Embrechts-Goldie-Veraverbeke formula shows that, in the classical Cramér-Lundberg risk model, the ruin probabilities satisfy $R(x, \infty ) \sim \rho ^{ - 1} \bar F_e (x)$ if the claim sizes are heavy-tailed, where Fe denotes the equilibrium distribution of the common d.f. F of the i.i.d. claims, ? is the safety loading coefficient of the model and the limit process is for x → ∞. In this paper we obtain a related local asymptotic relationship for the ruin probabilities. In doing this we establish two lemmas regarding the n-fold convolution of subexponential equilibrium distributions, which are of significance on their own right.

The compound Poisson risk model with dependence under a multi-layer dividend strategy

In this paper, a compound Poisson risk model with time-dependent claims is studiedunder a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber- Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let （X, Xk ： k ≥ 1） be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 〉 0.Let τ be a nonnegative integer-valued random variable, independent of {X, Xk ： k ≥ 1}. In this paper, the authors obtain the necessary and sufficient conditions for the random sums Sτ=∑n=1^τ Xn to have a consistently varying tail when the random number τ has a heavier tail than the summands, i.e.,P（X〉x）/P（τ〉x）→0 as x →∞.

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1, X2,... be a sequence of dependent and heavy-tailed random variables with distributions F1, F2,.. on （-∞,∞）, and let T be a nonnegative integer-valued random variable independent of the sequence {Xk, k 〉 1}. In this framework, the asymptotic behavior of the tail probabilities of the quantities Sn = fi Xk and S（n） =∑ k=1 n 〉 1, and their randomized versions ST and S（τ） are studied. Some risk theory are presented. max Sk for 1〈k〈n applications to the

On tail behavior of nonlinear autoregressive functional conditional heteroscedastic model with heavy-tailed innovations

We study the tail probability of the stationary distribution of nonparametric nonlinear autoregressive functional conditional heteroscedastic (NARFCH) model with heavytailed innovations. Our result shows that the tail of the stationary marginal distribution of an NARFCH series is heavily dependent on its conditional variance. When the innovations are heavy-tailed, the tail of the stationary marginal distribution of the series will become heavier or thinner than that of its innovations. We give some specific formulas to show how the increment or decrement of tail heaviness depends on the assumption on the conditional variance function. Some examples are given.

A time fractional model to represent rainfall process

This paper deals with a stochastic representation of the rainfall process. The analysis of a rainfall time series shows that cumulative representation of a rainfall time series can be modeled as a non-Gaussian random walk with a log-normal jump distribution and a time-waiting distribution following a tempered a-stable probability law. Based on the random walk model, a fractional Fokker-Planck equation （FFPE） with tempered a-stable waiting times was obtained. Through the comparison of observed data and simulated results from the random walk model and FFPE model with tempered a-stable waiting times, it can be concluded that the behavior of the rainfall process is globally reproduced, and the FFPE model with tempered a-stable waiting times is more efficient in reproducing the observed behavior.

Double-Penalized Quantile Regression in Partially Linear Models

In this paper, we propose the double-penalized quantile regression estimators in partially linear models. An iterative algorithm is proposed for solving the proposed optimization problem. Some numerical examples illustrate that the finite sample performances of proposed method perform better than the least squares based method with regard to the non-causal selection rate (NSR) and the median of model error (MME) when the error distribution is heavy-tail. Finally, we apply the proposed methodology to analyze the ragweed pollen level dataset.

Large Deviations for a Generalized Compound Renewal Risk Model

This paper extends the ordinary renewal risk model to the case where the premium income process,based on a renewal counting process,is no longer a linear function;and the total claim amount process is described by a compound renewal process.For this realistic risk model,the large deviations for the claim surplus process is investigated.