Specific Dependence Structure Research Articles

Asymptotic analysis of a dynamic systemic risk measure in a renewal risk model

In the context of insurance, we propose a dynamic systemic risk measure based on a multi-dimensional renewal risk model to describe the instant expected shortfall of an insurer at the moment when one of its business lines suffers crisis (i.e., deficit). The asymptotic behavior of the measure is studied in both the asymptotic independence and asymptotic dependence cases, and some asymptotic formulas with the uniformity in the whole time horizon are derived when the claim size distributions belong to the class of regular variation. The obtained results do not depend on specific setups of the renewal claim-number process, and they are also insensitive to specific dependence structures in the asymptotic independence case. These facts make our results concise in form and flexible in application. More interestingly, our results for the corresponding discounted version of the measure have nothing to do with the time variable. Hence, the discounted version of the measure can be regarded approximately as a static (i.e., time-invariant) risk measure of the dynamic system.

Analysis of multivariate non-gaussian functional data: A semiparametric latent process approach

Commonly assumed for multivariate functional regression models are normality and structural dependence, which, however, may not hold in practice. To relax these restrictions, we propose a new semiparametric transformation latent process functional regression model for multivariate functional data. Our model does not require normality assumptions or any specific dependence structures among multivariate response curves or intra-individual variability across time. We propose a combined likelihood- and estimating equation-based method to estimate parameters, transformation functions and covariance structures. We establish theoretical properties, including n−consistency and asymptotic normality, for the proposed estimators. The utility of the method is illustrated via extensive simulations and analyses of an elderly cognitive evolution dataset, which yield a better fit than the other competing methods and some interesting findings.

Tails of the Moments for Sums with Dominatedly Varying Random Summands

The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and Snξ=ξ1+…+ξn is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations ?Eξiα?{ξi>x} with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework.

Systemic risk allocation using the asymptotic marginal expected shortfall

This paper defines asymptotic marginal expected shortfall (AMES) for banks within a financial system and provides corresponding estimation method based on multivariate extreme value theory. The estimation method does not assume a specific dependence structure among bank equity returns. Both theoretical AMES and the estimator possess additive property and thus can serve as a tool to allocate system-wide risk to individual institutions. We apply the AMES to 30 global systemically important financial institutions (G-SIFIs). We show that the AMES outperforms the MES in predicting extreme losses during extreme systemic events. By taking the AMES as the reference point for allocating systemic risk to individual institutions, we show that an allocation according to simple bank characteristics such as size and individual risk can be imperfect. The allocation unfairness of individual risk or size across all the G-SIFIs has increased since 2008.

The nonparametric Behrens-Fisher problem in partially complete clustered data.

In randomized trials or observational studies involving clustered units, the assumption of independence within clusters is not practical. Existing parametric or semiparametric methods assume specific dependence structures within a cluster. Furthermore, parametric model assumptions may not even be realistic when data are measured in a nonmetric scale as commonly happens, for example, in quality-of-life outcomes. In this paper, nonparametric effect-size measures for clustered data that allow meaningful and interpretable probabilistic comparisons of treatments or intervention programs will be introduced. The dependence among observations within a cluster can be arbitrary. Point estimators along with their asymptotic properties for computing confidence intervals and performing hypothesis test will be discussed. Small sample approximations that retain some of the optimal asymptotic behaviors will be presented. In our setup, some clusters may involve observations coming from both intervention groups (referred to as complete clusters), while others may contain observations from one group only (referred to as incomplete clusters). In deriving the asymptotic theories, we do not impose any relation in the rate of divergence of the numbers of complete and incomplete clusters. Simulations show favorable performance of the methods for arbitrary combinations of complete and incomplete clusters. The developed nonparametric methods are illustrated using data from a randomized trial of indoor wood smoke reduction to improve asthma symptoms and a cluster-randomized trial for smokingcessation.

Estimation of the false discovery proportion with unknown dependence.

Large-scale multiple testing with correlated test statistics arises frequently in many scientific research. Incorporating correlation information in approximating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided an accurate approximation of False Discovery Proportion (FDP) under arbitrary dependence structure and some sparsity assumption. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before approximating FDP. The estimation accuracy can greatly affect FDP approximation. In the current paper, we aim to theoretically study the impact of unknown dependence on the testing procedure and establish a general framework such that FDP can be well approximated. The impacts of unknown dependence on approximating FDP are in the following two major aspects: through estimating eigenvalues/eigenvectors and through estimating marginal variances. To address the challenges in these two aspects, we firstly develop general requirements on estimates of eigenvalues and eigenvectors for a good approximation of FDP. We then give conditions on the structures of covariance matrices that satisfy such requirements. Such dependence structures include banded/sparse covariance matrices and (conditional) sparse precision matrices. Within this framework, we also consider a special example to illustrate our method where data are sampled from an approximate factor model, which encompasses most practical situations. We provide a good approximation of FDP via exploiting this specific dependence structure. The results are further generalized to the situation where the multivariate normality assumption is relaxed. Our results are demonstrated by simulation studies and some real data applications.

Precise large deviations of aggregate claims in a compound size-dependent renewal risk model

In this paper, we introduce a compound size-dependent renewal risk model driven by two sequences of random sources. The individual claim sizes and their inter-arrival times form a sequence of independent and identically distributed random pairs with each pair obeying a specific dependence structure. The numbers of claims caused by individual events form another sequence of independent and identically distributed positive integer-valued random variables, independent of the random pairs above. Precise large deviations of aggregate claims for the compound size-dependent renewal risk model are investigated in the case of dominatedly varying claim sizes.

Root-[formula omitted] consistent density estimation in GARCH models

We consider a new nonparametric estimator of the stationary density of the logarithm of the volatility of the GARCH(1,1) model. This problem is particularly challenging since this density is still unknown, even in cases where the model parameters are given. Although the volatility variables are only observed with multiplicative independent innovation errors with unknown density, we manage to construct a nonparametric procedure which estimates the log volatility density consistently. By carefully exploiting the specific GARCH dependence structure of the data, our iterative procedure even attains the striking parametric root-T convergence rate. As a by-product of our main results, we also derive new smoothness properties of the stationary density. Using numerical simulations, we illustrate the performance of our estimator, and we provide an application to financial data.

A new test procedure for the choice of dependence structure in risk measurement: application to the US and UK stock market indices

ABSTRACTThe choice of an appropriate dependence structure in modelling multivariate risks is an important issue because different tail structure embedded in copula leads to a different capital requirement for the institution. We present how to select a well-specified dependence structure to given application data. Using a simple simulation technique, we develop a statistical test to assess the adequacy of a specific dependence structure. We examine the sensitivity of risk estimates to the choice of copulas using the S&P 500 and FTSE 100 stock indices.

Newsvendor decisions under supply uncertainty

We analyse the impact of supply uncertainty on newsvendor decisions. First, we derive a solution for a newsvendor facing stochastic supply yield, in addition to stochastic demand. While earlier research has considered independent uncertainties, we derive the optimal order quantity for interdependent demand and supply and provide a closed-form solution for a specific copula-based dependence structure. This allows us to give insights into how dependence impacts the newsvendor’s decision, profit and risk level. In addition to the theory, we present experimental results that show how difficult newsvendor decisions under supply uncertainty are for human subjects. In our experiment, the control group replicated a well-known newsvendor experiment, whereas the test group faced additional supply yield uncertainty. Comparison of these results shows that under low-profit condition, subjects are able to incorporate supply uncertainty quite well in their decisions. Under high-profit condition, the deviation from the optimum is much more significant. We discuss this asymmetry and also propose some ways to improve newsvendor decision-making.

A Simple Density-Based Empirical Likelihood Ratio Test for Independence

We develop a novel nonparametric likelihood ratio test for independence between two random variables using a technique that is free of the common constraints of defining a given set of specific dependence structures. Our methodology revolves around an exact density-based empirical likelihood ratio test statistic that approximates in a distribution-free fashion the corresponding most powerful parametric likelihood ratio test. We demonstrate that the proposed test is very powerful in detecting general structures of dependence between two random variables, including nonlinear and/or random-effect dependence structures. An extensive Monte Carlo study confirms that the proposed test is superior to the classical nonparametric procedures across a variety of settings. The real-world applicability of the proposed test is illustrated using data from a study of biomarkers associated with myocardial infarction. Supplementary materials for this article are available online.

Systemic Risk Allocation for Systems with a Small Number of Banks

This paper provides a new estimation method for the marginal expected shortfall (MES) based on multivariate extreme value theory. In contrast to previous studies, the method does not assume specific dependence structure among bank equity returns and is applicable to both large and small systems. Furthermore, our MES estimator inherits the theoretical additive property. Thus, it serves as a tool to allocate systemic risk. We apply the proposed method to 29 global systemically important financial institutions (G-SIFIs) to evaluate the cross sections and dynamics of the systemic risk allocation. We show that allocating systemic risk according to either size or individual risk is imperfect and can be unfair. Between the allocation with respect to individual risk and that with respect to size, the former is less unfair. On the time dimension, both allocation fairness across all the G-SIFIs has decreased since 2008.

Positive quadrant dependence testing and constrained copula estimation

AbstractPositive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada

Discrete-Time Risk Models Based on Time Series for Count Random Variables

AbstractIn this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

On a risk model with stochastic premiums income and dependence between income and loss

Labbé and Sendova (2009) [9] consider a compound Poisson risk model with stochastic premiums income. In this paper, we extend their model by assuming that there exists a specific dependence structure among the claim sizes, interclaim times and premium sizes. Assume that the distributions of the premium sizes and interclaim times are controlled by the claim sizes. When the individual premium sizes are exponentially distributed, the Laplace transforms and defective renewal equations for the (Gerber–Shiu) discounted penalty functions are obtained. When the individual premium sizes have rational Laplace transforms, we show that the Laplace transforms for the discounted penalty functions can also be obtained.

Classification of slice-based VBR video traffic and estimation of link loss by exceedance

Classification of a video stream is an essential preliminary step to estimate the bit loss when the video stream is transmitted over a communication network . In this paper, we classify the video frames by the average frame size and estimate the bit loss for each class when the bitrate exceeds the capacity of the bottleneck link . The video stream under study is encoded using the explicit slice-based H.264/AVC encoding scheme. This scheme reduces the burstiness of regular H.264/AVC encoded video by removing the traditional GOP structure. Instead, a repetitive combination of intracoded and predicted slices is employed, thereby introducing a specific dependence structure in the video data. We consider a bufferless model of the communication system and evaluate the channel capacity required to give a maximum allowed loss rate for each class. Due to the high variability, non-stationarity and non-homogeneity of the underlying video data, the obtained classes are checked regarding the dependence and distribution structure of the data. The high quantiles of the losses are estimated for each class.

Solution Set Characterization of Linear Interval Systems with a Specific Dependence Structure

This is a contribution to solvability of linear interval equations and inequalities. In interval analysis we usually suppose that values from different intervals are mutually independent. This assumption can be sometimes too restrictive. In this article we derive extensions of Oettli-Prager theorem and Gerlach theorem for the case where there is a simple dependence structure between coefficients of an interval system. The dependence is given by equality of two submatrices of the constraint matrix.

Extreme dependence of multivariate catastrophic losses

Natural catastrophes cause insurance losses in several different lines of business. An approach to modelling the dependence in loss severities is to assume that they are related to the intensity of the natural disaster. In this paper we introduce a factor model and investigate the extreme dependence. We derive a specific extreme dependence structure when considering an heavy-tailed intensity. Estimation procedures are presented and their moderate sample properties are compared in a simulation study. We also motivate our approach by an illustrative example from storm insurance.

Exponential Behavior in the Presence of Dependence in Risk Theory

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.

Exponential Behavior in the Presence of Dependence in Risk Theory

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.