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.

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.

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.

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.

Median Unbiased Estimation of Bivariate Predictive Regression Models with Heavy-tailed or Heteroscedastic Errors

In this paper, we consider median unbiased estimation of bivariate predictive regression models with non-normal, heavy-tailed or heteroscedastic errors. We construct confidence intervals and median unbiased estimator for the parameter of interest. We show that the proposed estimator has better predictive potential than the usual least squares estimator via simulation. An empirical application to finance is given. And a possible extension of the estimation procedure to cointegration models is also described.

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.

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.

LARGE DEVIATIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES WITH DOMINATEDLY VARYING TAILS

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S（t）-ES（t）=N（t）∑i=1Xi-E（N（t）∑i=1Xi）,t≥0 are proved, where {N（t）;t ≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N（t）; t ≥ 0}. These results extend and improve some known conclusions.

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.

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.

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.

Optimal Rapid Restart of Heuristic Methods of NP Hard Problems

Many heuristic search methods exhibit a remarkable variability in the time required to solve some particular problem instances. Their cost distributions are often heavy-tailed. It has been demonstrated that, in most cases, rapid restart (RR) method can prominently suppress the heavy-tailed nature of the instances and improve computation efficiency. However, it is usually time-consuming to check whether an algorithm on a specific instance is heavy-tailed or not. Moreover, if the heavy-tailed distribution is confirmed and the RR method is relevant, an optimal RR threshold should be chosen to facilitate the RR mechanism. In this paper, an approximate approach is proposed to quickly check whether an algorithm on a specific instance is heavy-tailed or not. The method is realized by means of calculating the maximal Lyapunov exponent of its generic running trace. Then a statistical formula to estimate the optimal RR threshold is educed. The method is based on common nonparametric estimation, e.g., Kernel estimation. Two heuristic methods are selected to verify our method. The experimental results are consistent with the theoretical consideration perfectly.

Nearly nonstationary processes under infinite variance GARCH noises

Let Y_(t) be an autoregressive process with order one,i.e.,Y_(t)=μ+ϕnY_(t-1)+εt,where fεtg is a heavy tailed general GARCH noise with tail indexα.Letϕn be the least squares estimator(LSE)ofϕn.Forμ=0 andα<2,it is shown by Zhang and Ling(2015)thatϕn is inconsistent when Y_(t) is stationary(i.e.,ϕn.,ϕ<1),however,Chan and Zhang(2010)showed thatϕn is still consistent with convergence rate n when Y_(t) is a unit-root process(i.e.,ϕn=1)and fεtg is a GARCH(1,1)noise.There is a gap between the stationary and nonstationary cases.In this paper,two important issues will be considered:(1)what about the nearly unit root case?(2)When canϕbe estimated consistently by the LSE?We show that whenϕn=1-c/n,then bϕn converges to a functional of stable process with convergence rate n.Further,we show that if limn!1 kn(1-ϕn)=c for a positive constant c,then kn(ϕn-ϕ)converges to a functional of two stable variables with tail indexα/2,which means thatϕn can be estimated consistently only when kn!1.

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.

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.

Functional Kernel Estimation of the Conditional Extreme Quantile under Random Right Censoring

The study of estimation of conditional extreme quantile in incomplete data frameworks is of growing interest. Specially, the estimation of the extreme value index in a censorship framework has been the purpose of many investigations when finite dimension covariate information has been considered. In this paper, the estimation of the conditional extreme quantile of a heavy-tailed distribution is discussed when some functional random covariate (i.e. valued in some infinite-dimensional space) information is available and the scalar response variable is right-censored. A Weissman-type estimator of conditional extreme quantiles is proposed and its asymptotic normality is established under mild assumptions. A simulation study is conducted to assess the finite-sample behavior of the proposed estimator and a comparison with two simple estimations strategies is provided.

High-dimensional robust inference for censored linear models

Due to the direct statistical interpretation,censored linear regression offers a valuable complement to the Cox proportional hazards regression in survival analysis.We propose a rank-based high-dimensional inference for censored linear regression without imposing any moment condition on the model error.We develop a theory of the high-dimensional U-statistic,circumvent challenges stemming from the non-smoothness of the loss function,and establish the convergence rate of the regularized estimator and the asymptotic normality of the resulting de-biased estimator as well as the consistency of the asymptotic variance estimation.As censoring can be viewed as a way of trimming,it strengthens the robustness of the rank-based high-dimensional inference,particularly for the heavy-tailed model error or the outlier in the presence of the response.We evaluate the finite-sample performance of the proposed method via extensive simulation studies and demonstrate its utility by applying it to a subcohort study from The Cancer Genome Atlas(TCGA).

Asymptotics for the joint tail probability of bidimensional randomly weighted sums with applications to insurance

This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectively,are real-valued,dependent and heavy-tailed,while the random weights{Θi,θi;i∈N}are nonnegative and arbitrarily dependent,but the three sequences{X_(i);i∈N},{Y_(i);i∈N}and{Θ_(i),θ_(i);i∈N}are mutually independent.Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights,we establish some(uniformly)asymptotic formulas for the joint tail probability of the two randomly weighted sums,expressing the insensitivity with respect to the underlying weak dependence structures.As applications,we consider both discrete-time and continuous-time insurance risk models,and obtain some asymptotic results for ruin probabilities.

Uniform Asymptotics for Finite-Time Ruin Probabilities of Risk Models with Non-Stationary Arrivals and Strongly Subexponential Claim Sizes

This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process.Under the assumption that the claim-number process satisfies the large deviations principle,the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes.Further,as an application of the result of onedimensional risk model,we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.