Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n ? 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.

The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.     

Copula Density Estimation by Total Variation Penalized Likelihood

Copulas are full measures of dependence among random variables. They are increasingly popular among academics and practitioners in financial econometrics for modeling comovements between markets, risk factors, and other relevant variables. A copula's hidden dependence structure that couples a joint distribution with its marginals makes a parametric copula non-trivial. An approach to bivariate copula density estimation is introduced that is based on a penalized likelihood with a total variation penalty term. Adaptive choice of the amount of regularization is based on approximate Bayesian Information Criterion (BIC) type scores. Performance are evaluated through the Monte Carlo simulation.     

股票组合VaR估计：基于动态Copula方法

已有的使用动态时变Copula估计VaR的研究都仅限于考虑两个资产,对两个资产以上,Copula函数的参数过多,逐一设定参数的动态过程,将使模型复杂化,在计算上也不可行。为解决这一问题,文中使用条件Copula的概念,结合Engle的DCC方法,将椭球Copula的相关系数矩阵动态化,并将t-Copula的自由度设定为一动态过程的Logistic变换,由此得到的动态正态Copula和t-Copula可用于刻画两个以上资产相关结构的动态关系,进而可估计两个以上资产组合的VaR。最后,文章给出了一个经验应用。     

Transformation-Kernel Estimation of Copula Densities

ABSTRACT

The standard kernel estimator of copula densities suffers from boundary biases and inconsistency due to unbounded densities. Transforming the domain of estimation into an unbounded one remedies both problems, but also introduces an unbounded multiplier that may produce erratic boundary behaviors in the final density estimate. We propose an improved transformation-kernel estimator that employs a smooth tapering device to counter the undesirable influence of the multiplier. We establish the theoretical properties of the new estimator and its automatic higher-order improvement under Gaussian copulas. We present two practical methods of smoothing parameter selection. Extensive Monte Carlo simulations demonstrate the competence of the proposed estimator in terms of global and tail performance. Two real-world examples are provided. Supplementary materials for this article are available online.     

Mixtures of Mixtures Based on Multiresolution Analysis Theory

In this article, the approaches for exploiting mixtures of mixtures are expanded by using the Multiresolution family of probability density functions (MR pdf). The flexibility and the properties of local analysis of the MR pdf facilitate the location of subpopulations into a given population. In order to do this, two algorithms are provided.

The MR model is more flexible in adapting to the different subpopulations than the traditional mixtures. In addition, the problems of identification of mixtures distributions and the label-switching do not appear in the MR pdf context.     

基于分层阿基米德Copula的金融时间序列的相关性分析

与阿基米德copula相比,分层阿基米德copula(HAC)的结构更具一般性,而相比于椭圆型copula它的待估参数个数更少。用两阶段极大似然法来估计HAC函数,主要的步骤是先估计出每个分量的边际分布,以此为基础再估计copula函数。实证分析中,采取Clayton和Gumbel型的HAC分析四只股票价格序列之间的相关性。在得出HAC的结构和估计其参数之前,运用ARMA-GARCH过程消除了序列的自相关性和条件异方差。通过比较赤迟信息准则,认为完全嵌套的Gumbel型HAC能更好地刻画这种相关性。     

基于小波多分辨分析的协整建模理论与方法的扩展;

为讨论经济及金融变量的多尺度行为,描述变量之间在不同时间尺度上的长期均衡关系,将小波多分辨分析引入协整建模理论,提出多分辨协整和多分辨误差校正模型两个概念,给出相应建模方法,克服了传统的协整建模理论无法揭示蕴含在变量内部的多时间尺度信息的缺陷。多分辨协整建模能够更加细致地捕获经济或金融变量在不同时间尺度上的关系,对两大股指的实证研究也支持了这一点。     

MCMC方法下最优Copula的估计及选取

针对目前Copula函数在实际中的应用问题,介绍了一种基于马尔科夫链蒙特卡罗方法（MCMC）的Copula函数估计及选取方法,并将该方法与目前常用方法进行系统比较,最后对上证综合指数和深证成分指数进行了实证分析,结果体现了该法的有效性。     

基于Copula方法的极值洪水频率与风险分析

洪灾风险分析中的一项重要内容是关于洪水发生概率(频率)的估计。以湖南省四大水系中的四个站点50多年的年最高水位数据为基础,结合常用于灾害分析中的分布模型估计出每个站点的洪水频率分布,再利用Copula函数模型得到两两水系间水位协同变化的联合分布函数,进而估计每两条河流同时发生洪水灾害的概率。     

基于Copula函数的金融市场尾部相关性分析

在常规极大似然估计法中,Copula函数的参数估计受边缘分布函数拟和的影响较大,鉴于此,用基于秩的极大似然法估计Copula函数的参数,并结合常见的4类双参数非对称BBx—Copula函数,对民生银行和浦发银行这两只股票的尾部相关性进行实证分析,结果表明股票市场在低迷时期的尾部相关性高于活跃时期的尾部相关性。     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

基于Copula连接函数的数据挖掘关联算法的设计

现实中海量数据往往持续地产生,如何实现信息和知识的动态挖掘已成为人们关注的理论问题。根据数据集分批分步输入处理的思想,以Copula连接函数为理论基础,给出一种有效海量数据的关联分步测度算法,通过模拟实验验证了该算法的可行性,结果显示所设计的关联算法能显著提高关联效应测量的效率,并能有效地解决超海量数据关联效应的测度问题。     

Reliability Research of k-out-of-n: G Supply Chain System Based on Copula

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article.     

基于Copula理论的最优边缘分布模型构建

基于Copula理论,文章以Kolmogorov-Smirnov、Ljung-Box Q检验结果为择优标准,综合利用ARMA-GARCH族模型、极值理论以及非参数核密度函数构建四类常用边缘分布模型,以确定刻画全球重要股指收益率边缘分布的最优模型。结果表明,非参数ARMA-GARCH族-EVT模型对样本边缘分布的刻画最为准确。     

基于Copula方法的干散货运费子市场尾部相关性分析

通过分析Copula函数的尾部相关性来揭示干散货巴拿马运费市场和好望角运费市场的相关性。实证结果表明,基于时变相关的二元对称Joe-Clayton(SJC)Copula对干散货运费子市场尾部相关性的拟合效果最好,而且当市场下降时,两个子市场表现出强而剧烈的相关性;市场上升时,二者表现出弱而平稳的相关性,因此基于不同船型的航线组合投资在市场上升时的效果优于市场下降。     

基于藤Copula分组模型的金融市场风险度量研究

传统未分组的藤Copula模型可用于刻画金融资产间的相依性,但其存在将所有不同行业资产视为一个整体的问题。本文在充分考虑金融市场中各机构所属行业不同的基础上,提出了藤Copula分组模型,给出了该模型算法的具体步骤,并证明了算法的收敛性。最后通过返回检验方法,对比研究了藤Copula分组模型和未分组的藤Copula模型对银行业、证券业和保险业间VAR估计的精度差异,结果表明藤Copula分组模型的预测效果更准确且更有效。     

基于核密度估计的非线性时间序列聚类

本文研究的是时间序列的聚类问题。由于现实世界中时间序列多数是非线性的,而现有的时间序列聚类问题大都是基于线性时间序列模型进行聚类的,本文提出了可以用于非线性时间序列的聚类方法。以时间序列的二维核密度估计之间的相似性作为非线性时间序列的距离度量,该距离度量方式是一种非参数的距离度量方法,考虑到了时间序列自相关结构的差异,能够粗糙地识别时间序列形状和动态相关结构的相似性。与理论研究结果相一致,我们的模拟实验结果也验证了这种距离度量的有效性。     

On the Estimation of the Probability Density by Trigonometric Series

We present in this article an estimator based on a new orthogonal trigonometric series. We give its statistical properties (bias, variance, mean square error, and mean integrated square error) and the asymptotic properties (convergence of variance, convergence of the mean square error, convergence of the mean integrated square error, uniform convergence in probability, and the rate of convergence of the mean integrated square error). The comparison by simulation on a test density between the estimator obtained from a new trigonometric series with Fejer estimator also based on orthogonal trigonometric series, shows that our estimator is more performant in the sense of the mean integrated square error.     

Estimation of the Spectral Density with Assigned Risk

It is well known that adaptive sequential nonparametric estimation of differentiable functions with assigned mean integrated squared error and minimax expected stopping time is impossible. In other words, no sequential estimator can compete with an oracle estimator that knows how many derivatives an estimated curve has. Differentiable functions are typical in probability density and regression models but not in spectral density models, where considered functions are typically smoother. This paper shows that for a large class of spectral densities, which includes spectral densities of classical autoregressive moving average processes, an adaptive minimax sequential estimation with assigned mean integrated squared error is possible. Furthermore, a two‐stage sequential procedure is proposed, which is minimax and adaptive to smoothness of an underlying spectral density.     

Nonparametric Wavelet Regression Based on Biased Data

The estimation of the regression function in the biased nonparametric regression model is investigated. We propose and develop a new wavelet-based methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls.