Truncated regular vines in high dimensions with application to financial data

Using only bivariate copulas as building blocks, regular vine copulas constitute a flexible class of high‐dimensional dependency models. However, the flexibility comes along with an exponentially increasing complexity in larger dimensions. In order to counteract this problem, we propose using statistical model selection techniques to either truncate or simplify a regular vine copula. As a special case, we consider the simplification of a canonical vine copula using a multivariate copula as previously treated by Heinen & Valdesogo ( 2009 ) and Valdesogo ( 2009 ). We validate the proposed approaches by extensive simulation studies and use them to investigate a 19‐dimensional financial data set of Norwegian and international market variables. The Canadian Journal of Statistics 40: 68–85; 2012 © 2012 Statistical Society of Canada     

On the diversity score: a copula approach

ABSTRACT

For the rating process of Collateralized Debt Obligations', Moody's suggests the Diversity Score as a measure of diversification in the collateral pool. This measure is used in Moody's Binomial Expansion Technique to infer the probability of default and thus the expected Loss in the portfolio. In this paper, we examine the appropriateness of this approach to assess the reality of defaults using a copula approach and lower tail dependence.     

Bayesian model selection for D‐vine pair‐copula constructions

In recent years analyses of dependence structures using copulas have become more popular than the standard correlation analysis. Starting from Aas et al. ( 2009 ) regular vine pair‐copula constructions (PCCs) are considered the most flexible class of multivariate copulas. PCCs are involved objects but (conditional) independence present in data can simplify and reduce them significantly. In this paper the authors detect (conditional) independence in a particular vine PCC model based on bivariate t copulas by deriving and implementing a reversible jump Markov chain Monte Carlo algorithm. However, the methodology is general and can be extended to any regular vine PCC and to all known bivariate copula families. The proposed approach considers model selection and estimation problems for PCCs simultaneously. The effectiveness of the developed algorithm is shown in simulations and its usefulness is illustrated in two real data applications. The Canadian Journal of Statistics 39: 239–258; 2011 © 2011 Statistical Society of Canada     

Polynomial approximation of nonlinear regression functions

In the present paper a nonlinear regression function is approximated by a polynomial estimator according to the expectation of the quadratic L ²-distance as risk is given. For special experimental designs with repeating experimental points this estimator coincides with the estimator by the method of the reproducing kernel.

Considerations about the relation for the sample size and the degree of the approximation polynomial and about the quadratic mean are given.     

Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

Estimators based on trimmed Kendall’s tau in multivariate copula models

A common method of estimating the parameters of dependency in multivariate copula models is by maximum likelihood principle, termed as Inference From Marginals (IFM); see Joe (1997)  [13]. To avoid possible misspecification of the marginal distributions, some authors suggest rank-based procedures for estimating the parameters of dependency in a multivariate copula model. A standard approach for this problem is through maximization of the pseudolikelihood, as discussed in Genest et al. (1995)  [9] and Shih and Louis (1995)  [23]. Alternative estimators based on the inversion of two multivariate extensions of Kendall’s tau, due to Kendall and Babington Smith (1940)  [14] and Joe (1990)  [12], were used in Genest et al. (2011)  [10]. In the literature, dependency of data was considered in the whole data space. However, it may be better to divide the data set into two distinct sets, lower and higher than a threshold, and then evaluate the dependency parameters in these sets. In this way, we may have different dependency parameters in these sets which may shed additional light. For example, in drought analysis, precipitation and minimum temperature may be modeled using copulas in which case we can infer that dependency between precipitation and minimum temperature are severe when they are less than a certain threshold. In this paper, after introducing trimmed Kendall’s tau when such a threshold is imposed, we consider modeling dependency using it as a measure. Asymptotic distribution of trimmed Kendall’s tau is also investigated, and a test for the null hypothesis of equality between Kendall’s tau and trimmed Kendall’s tau is constructed. We can use this hypothesis testing procedure for testing the hypothesis that data are dependent before a threshold value and are independent after the threshold. An explicit form of the asymptotic distribution of trimmed Kendall’s tau and of the mentioned test statistic are also derived for some special families of copulas. Finally, the results of a simulation study and an illustrative example are provided.     

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     

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。     

Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components

By adding a resilience parameter to the scale model, a general distribution family called resilience-scale model is introduced including exponential, Weibull, generalized exponential, exponentiated Weibull and exponentiated Lomax distributions as special cases. This paper carries out stochastic comparisons on parallel and series systems with heterogeneous resilience-scaled components. On the one hand, it is shown that more heterogeneity among the resilience-scaled components of a parallel [series] system with an Archimedean [survival] copula leads to better [worse] performance in the sense of the usual stochastic order. On the other hand, the [reversed hazard] hazard rate order is established for two series [parallel] systems consisting of independent heterogeneous resilience-scaled components. The skewness and dispersiveness are also investigated for the lifetimes of two parallel systems consisting of independent heterogeneous and homogeneous [multiple-outlier] resilience-scaled components. Numerical examples are provided to illustrate the effectiveness of our theoretical findings. These results not only generalize and extend some known ones in the literature, but also provide guidance for engineers to assemble systems with higher reliability in practical situations.     

A bivariate geometric distribution allowing for positive or negative correlation

In this paper, we propose a new bivariate geometric model, derived by linking two univariate geometric distributions through a specific copula function, allowing for positive and negative correlations. Some properties of this joint distribution are presented and discussed, with particular reference to attainable correlations, conditional distributions, reliability concepts, and parameter estimation. A Monte Carlo simulation study empirically evaluates and compares the performance of the proposed estimators in terms of bias and standard error. Finally, in order to demonstrate its usefulness, the model is applied to a real data set.