Types Of Dependence Structures Research Articles

Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distributions whose tail is in the intermediate heavy-tailed regime, which includes Weibull and lognormal type tails. The study derives asymptotic approximations for tail events of random walks. Consequently, a full large deviations principle is obtained under, essentially, minimal assumptions. As an application, an optimisation method for a large class of Quota Share (QS) risk sharing schemes used in insurance and finance is obtained.

Multivariate Bayesian hierarchical Gaussian copula modeling of the non-stationary traffic conflict extremes for crash estimation

Recent studies have demonstrated that single conflict indicators represent only a fractional aspect of the severity of a traffic interaction. As such, integrating several conflict indicators in a unified model can improve conflict-based crash estimation. This study develops a multivariate Bayesian hierarchical Gaussian copula modeling approach, which comprises a multivariate Gaussian copula and a Bayesian hierarchical structure. The former has generalized extreme value marginals and captures multivariate dependence among several conflict indicators, while the latter combines traffic conflicts from different sites, incorporating several covariates and site-specific unobserved heterogeneity. The copula approach offers the flexibility in modeling multivariate structures including the selection of margins from various parametric families of univariate distribution and the construction of parametric copulas which satisfies different types of dependence structure. A model estimation approach for the multivariate Bayesian hierarchical Gaussian copula model is proposed and applied to estimate rear-end crashes from four signalized intersections in the city of Surrey, British Columbia. The modified time to collision (MTTC), post encroachment time (PET), and deceleration rate to avoid a crash (DRAC) were employed as conflict indicators. Three covariates including traffic volume, shock wave area, and platoon ratio were considered to account for non-stationarity in conflict extremes. The results show that in terms of crash estimation accuracy, the multivariate Bayesian hierarchical Gaussian copula model outperforms both the bivariate Bayesian hierarchical Gaussian copula models and the recently proposed multivariate Bayesian hierarchical model in which multiple traffic conflict indicators were combined by one dependence parameter.

The Finite-time Ruin Probability of a Discrete-time Risk Model with Subexponential and Dependent Insurance and Financial Risks

Consider a discrete-time risk model with insurance and financial risks in a stochastic economic environment. Assume that the insurance and financial risks form a sequence of independent and identically distributed random vectors with a generic random vector following a wide type of dependence structure. An asymptotic formula for the finite-time ruin probability with subexponential insurance risks is derived. In doing so, the subexponentiality of the product of two dependent random variables is investigated simultaneously.

Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation

ABSTRACTLet Xk, 1 ⩽ k ⩽ n, be n real-valued random variables, and θk, 1 ⩽ k ⩽ n, be another n non negative not-degenerate at zero random variables. Assume that random pairs (X1, θ1), …, (Xn, θn) are mutually independent, while each pair (Xk, θk) follows a wide type of dependence structure. Consider the randomly weighted sum Sθn = ∑k = 1nθkXk. In this paper, the tail asymptotics for Sθn in the case where Xk, 1 ⩽ k ⩽ n, belong to some heavy-tailed subclasses are firstly investigated. Then, as an application, we consider the tail behavior of the conditional tail expectation as q↑1, where . Under some technical conditions, the asymptotic estimate for the right tail of conditional tail expectation is also derived. The obtained results extend some existing ones in the literature.

Some novel results on pairwise quasi-asymptotical independence with applications to risk theory

ABSTRACTIn this article we obtain some novel results on pairwise quasi-asymptotically independent (pQAI) random variables. Concretely speaking, let X1, …, Xn be n real-valued pQAI random variables, and W1, …, Wn be another n non negative and arbitrarily dependent random variables, but independent of X1, …, Xn. Under some mild conditions, we prove that W1X1, …, WnXn are still pQAI as well. Our result is in a general setting whether the primary random variables X1, …, Xn are heavy-tailed or not. Finally, a special case of above result is applied to risk theory for investigating the finite-time ruin probability for a discrete-time risk model with a wide type of dependence structure.

Construction and identification of a D-Vine model applied to the probability distribution of modal parameters in structural dynamics

This study investigates the construction and identification of the probability distribution of random modal parameters (natural frequencies and effective parameters) in structural dynamics. As these parameters present various types of dependence structures, the retained approach is based on pair copula construction (PCC). A literature review leads us to choose a D-Vine model for the construction of modal parameters probability distributions. Identification of this model is based on likelihood maximization which makes it sensitive to the dimension of the distribution, namely the number of considered modes in our context. To this respect, a mode selection preprocessing step is proposed. It allows the selection of the relevant random modes for a given transfer function. The second point, addressed in this study, concerns the choice of the D-Vine model. Indeed, D-Vine model is not uniquely defined. Two strategies are proposed and compared. The first one is based on the context of the study whereas the second one is purely based on statistical considerations. Finally, the proposed approaches are numerically studied and compared with respect to their capabilities, first in the identification of the probability distribution of random modal parameters and second in the estimation of the 99% quantiles of some transfer functions.

Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks

This contribution focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account and they are equipped with a wide type of dependence structure. We derive precise asymptotic formulas for the ruin probabilities when the insurance risk has a dominatedly varying tail. In the special case of regular variation, the corresponding formula is proved to be uniform for the time horizon.

A SIMPLE EXTENSION OF COMONOTONICITY:FROM INDEPENDENCE TO COMONOTONICITY

Comonotonicity has become popular in actuarial science and finance.The notion of upper comonotonicity has recently been proposed.Using distributional representation we provide a unified method to extend the notion of comonotonicity further to lower comonotonicity,lower and upper comonotonicity,and interval comonotonicity.Numerical illustrations are provided to make a comparison among the six types of dependence structure:Independence,upper comonotonicity,lower comonotonicity,lower and upper comonotonicity,interval comonotonicity,comonotonicity.The numerical results are related to the sum of uniform(0,1) random variables,for which we obtain the explicit formula for the density function of the sum of two random variables in every case.For higher dimension,it becomes complicated to find the corresponding explicit formulas.

Modeling the co-movements between crude oil and refined petroleum markets

In this paper we investigate two types of asymmetries, i.e., the asymmetry in the lower and upper tail dependences and the asymmetry in the propagation of crisis (bubble), between crude oil market and refined petroleum markets based on copula models. Thirteen copula models with different types of dependence structures and time-varying dependence parameters are considered. We find that in general the MALM copula fits our sample data best based on AIC criterion. We find that lower and upper tail dependences are both positive indicating that crude oil and refined product markets tend to move together. Using the asymmetric copulas, we find asymmetry in tail dependence between crude oil and heating oil returns and that between crude oil and jet fuel returns. Interestingly, upper tail dependence is significantly greater than the lower tail dependence for the pre-crisis period, while the result is reversed for the post-crisis period. Finally, although our data prefers the nonexchangeable MALM copula, the asymmetry in the propagation of crisis (bubble) between crude oil and refined product returns is very weak.

On piecewise polynomial regression under general dependence conditions, with an application to calcium-imaging data

Motivated by the analysis of glomerular time series extracted from calcium-imaging data, asymptotic theory for piecewise polynomial and spline regression with partially free knots and residuals exhibiting three types of dependence structures (long memory, short memory and anti-persistence) is considered. Unified formulas based on fractional calculus are derived for subordinated residual processes in the domain of attraction of a Hermite process. The results are applied to testing for the effect of a neurotransmitter on the response of olfactory neurons in honeybees to odorant stimuli.

Extensions of the notion of overall comonotonicity to partial comonotonicity

The overall comonotonicity has become popular in actuarial science and finance over the last decade. As a further step, the notion of upper comonotonicity has recently been proposed. Using the technique of distributional representation we provide a unified method to extend the notion of comonotonicity further to lower comonotonicity, tail comonotonicity, and interval comonotonicity respectively. Numerical illustrations are provided to make a comparison among these different types of dependence structures. The numerical results can be explained to some extent by the sum of uniform (0,1) random variables, for which we obtain explicit formulae for the probability density functions of the sum of two random variables in partial comonotonicity cases. For higher dimension cases, it becomes complicated to find the corresponding explicit formulae.

Student processes

Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

Student processes

Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.

Modeling multivariate survival data by a semiparametric random effects proportional odds model.

In this article, the focus is on the analysis of multivariate survival time data with various types of dependence structures. Examples of multivariate survival data include clustered data and repeated measurements from the same subject, such as the interrecurrence times of cancer tumors. A random effect semiparametric proportional odds model is proposed as an alternative to the proportional hazards model. The distribution of the random effects is assumed to be multivariate normal and the random effect is assumed to act additively to the baseline log-odds function. This class of models, which includes the usual shared random effects model, the additive variance components model, and the dynamic random effects model as special cases, is highly flexible and is capable of modeling a wide range of multivariate survival data. A unified estimation procedure is proposed to estimate the regression and dependence parameters simultaneously by means of a marginal-likelihood approach. Unlike the fully parametric case, the regression parameter estimate is not sensitive to the choice of correlation structure of the random effects. The marginal likelihood is approximated by the Monte Carlo method. Simulation studies are carried out to investigate the performance of the proposed method. The proposed method is applied to two well-known data sets, including clustered data and recurrent event times data.

Processes of normal inverse Gaussian type

With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.