Examination and visualisation of the simplifying assumption for vine copulas in three dimensions

Vine copulas are a highly flexible class of dependence models, which are based on the decomposition of the density into bivariate building blocks. For applications one usually makes the simplifying assumption that copulas of conditional distributions are independent of the variables on which they are conditioned. However this assumption has been criticised for being too restrictive. We examine both simplified and non‐simplified vine copulas in three dimensions and investigate conceptual differences. We show and compare contour surfaces of three‐dimensional vine copula models, which prove to be much more informative than the contour lines of the bivariate marginals. Our investigation shows that non‐simplified vine copulas can exhibit arbitrarily irregular shapes, whereas simplified vine copulas appear to be smooth extrapolations of their bivariate margins to three dimensions. In addition to a variety of constructed examples, we also investigate a three‐dimensional subset of the well‐known uranium data set and visually detect the fact that a non‐simplified vine copula is necessary to capture its complex dependence structure.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

Estimation of the Association for Bivariate Interval-censored Failure Time Data

Abstract.  Multivariate failure time data frequently occur in medical studies and the dependence or association among survival variables is often of interest ( Biometrics , 51 , 1995, 1384; Stat. Med. , 18 , 1999, 3101; Biometrika , 87 , 2000, 879; J. Roy. Statist. Soc. Ser. B , 65 , 2003, 257). We study the problem of estimating the association between two related survival variables when they follow a copula model and only bivariate interval-censored failure time data are available. For the problem, a two-stage estimation procedure is proposed and the asymptotic properties of the proposed estimator are established. Simulation studies are conducted to assess the finite sample properties of the presented estimate and the results suggest that the method works well for practical situations. An example from an acquired immunodeficiency syndrome clinical trial is discussed.     

股指期货与现货指数收益率序列相关性研究——基于Copula函数的实证分析

沪深300股指期货推出后,其与沪深300指数的关系就引起投资者和研究者的关注。以沪深300指数和沪深300股指期货的日收益率数据为基础,运用Copula函数建立Copula-GARCH(1,1)-GED模型对两者进行相关性分析,结果表明:沪深300指数与股指期货收益率序列之间相关程度非常高,而通过比较秩相关系数的拟合情况,二元正态Copula函数更接近实际情况;在平方欧式距离的标准下,二元t-Copula模型能够更好地描述沪深300指数与沪深300股指期货日收益率序列的相关结构;两序列的尾部相关程度非常高,表明当股票市场大幅度波动时,沪深300指数与沪深300股指期货的相关程度显著提高。     

心理账户交互作用下证券投资组合风险度量模型研究

考虑证券投资组合决策中心理账户因素, 基于Copula函数和信息熵原理, 构建了心理账户交互作用下证券投资组合Copula-IE风险度量模型, 改进了传统行为证券组合风险度量模型, 提出了新的行为证券组合风险度量优化模型。通过对该模型分析可知:投资者的投资预期水平越高, 所造成的行为决策风险越高;投资者心理账户之间的替代性越强或越弱, 均使得投资组合的风险增强。     

中国创业板和主板市场之间的相关结构分析——基于Copula函数的实证研究

了解创业板与主板市场间的关系,对于理解两个市场的互动关系以及证券组合的设计和风险评估至关重要。文章利用创业板指数、沪深300指数数据,借助ARMA-GARCH-Normal-Copula和ARMA-GARCH-T-Copula模型分析了创业板和主板之间的相关结构。研究表明:创业板市场的波动要大于主板市场,创业板和主板存在一定的正相关关系,并且存在非对称的尾部结构;在尾部(上尾和下尾)上,创业板对主板的变化反应比主板对创业板的变化反应更强。     

Optimal Weight Functions for Marginal Proportional Hazards Analysis of Clustered Failure Time Data

The choice of weights in estimating equations for multivariate survival data is considered. Specifically, we consider families of weight functions which are constant on fixed time intervals, including the special case of time-constant weights. For a fixed set of time intervals, the optimal weights are identified as the solution to a system of linear equations. The optimal weights are computed for several scenarios. It is found that for the scenarios examined, the gains in efficiency using the optimal weights are quite small relative to simpler approaches except under extreme dependence, and that a simple estimator of an exchangeable approximation to the weights also performs well.     

Statistical Modeling of Temporal Dependence in Financial Data via a Copula Function

In financial analysis it is useful to study the dependence between two or more time series as well as the temporal dependence in a univariate time series. This article is concerned with the statistical modeling of the dependence structure in a univariate financial time series using the concept of copula. We treat the series of financial returns as a first order Markov process. The Archimedean two-parameter BB7 copula is adopted to describe the underlying dependence structure between two consecutive returns, while the log-Dagum distribution is employed to model the margins marked by skewness and kurtosis. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. Furthermore, we apply the model to the daily returns of four stocks and, finally, we illustrate how its fitting to data can be improved when the dependence between consecutive returns is described through a copula function.     

A copula-based approach for estimating the survival functions of two alternating recurrent events

In this paper, we study the survival times of alternately occurring events. The dependence between the times to the two events is modelled through the Archimedean copula, while the dependence over the recurring cycles is modelled through a functional relationship of the distribution parameters. Taking account of appropriate censoring that may be present in the data, the model parameters are estimated using the maximum likelihood method. The standard errors of the estimators are then derived and confidence belts for the survival functions constructed. Methods for choosing the appropriate copula are also discussed. The results are illustrated through a clinical trial data on patients suffering from cystic fibrosis. A simulation study is also done to corroborate the results.