Hazard Estimation With Bivariate Survival Data and Copula Density Estimation

Bivariate survival function can be expressed as the composition of marginal survival functions and a bivariate copula and, consequently, one may estimate bivariate hazard functions via marginal hazard estimation and copula density estimation. Leveraging on earlier developments on penalized likelihood density and hazard estimation, a nonparametric approach to bivariate hazard estimation is being explored in this article. The new ingredient here is the nonparametric estimation of copula density, a subject of interest by itself, and to accommodate survival data one needs to allow for censoring and truncation in the setting. A simple copularization process is implemented to convert density estimates into copula densities, and a cross-validation scheme is devised for density estimation under censoring and truncation. Empirical performances of the techniques are investigated through simulation studies, and potential applications are illustrated using real-data examples and open-source software.     

A note on bootstrap approximations for the empirical copula process

It is well known that the empirical copula process converges weakly to a centered Gaussian field. Because the covariance structure of the limiting process depends on the partial derivatives of the unknown copula several bootstrap approximations for the empirical copula process have been proposed in the literature. We present a brief review of these procedures. Because some of these procedures also require the estimation of the derivatives of the unknown copula we propose an alternative approach which circumvents this problem. Finally a simulation study is presented in order to compare the different bootstrap approximations for the empirical copula process.     

Estimation Procedures for a Semiparametric Family of Bivariate Copulas

This article proposes simple estimation methods dedicated to a semiparametric family of bivariate copulas. These copulas can be simply estimated through the estimation of their univariate generating function. We use this result to estimate the associated measures of association as well as the high probability regions of the copula. These procedures are illustrated using both simulations and real data.     

Efficient MCMC estimation of some elliptical copula regression models through scale mixtures of normals

This paper proposes an efficient estimation method for some elliptical copula regression models by expressing both copula density and marginal density functions as scale mixtures of normals (SMN). Implementing these models using the SMN is novel and allows efficient estimation via Bayesian methods. An innovative algorithm for the case of complex semicontinuous margins is also presented. We utilize the facts that copulas are invariant to the location and scale of the margins; all elliptical distributions have the same correlation structure; and some densities can be represented by the SMN. Two simulation studies, one on continuous margins and the other on semicontinuous margins, highlight the favorable performance of the proposed methods. Two empirical studies, one on the US excess returns and one on the Thai wage earnings, further illustrate the applicability of the proposals.     

Rank-based inference tools for copula regression,with property and casualty insurance applications

Rank-based procedures are commonly used for inference in copula models for continuous responses whose behavior does not depend on covariates. This paper describes how these procedures can be adapted to the broader framework in which (possibly non-linear) regression models for the marginal responses are linked by a copula that does not depend on covariates. The validity of many of these techniques can be derived from the asymptotic equivalence between the classical empirical copula process and its analog based on suitable residuals from the marginal models. Moment-based parameter estimation and copula goodness-of-fit tests are shown to remain valid under weak conditions on the marginal error term distributions, even when the residual-based empirical copula process fails to converge weakly. The performance of these procedures is evaluated through simulation in the context of two general insurance applications: micro-level multivariate insurance claims, and dependent loss triangles.     

Moment-recovered approximations of multivariate distributions: The Laplace transform inversion

The moment-recovered approximations of multivariate distributions are suggested. This method is natural in certain incomplete models where moments of the underlying distribution can be estimated from a sample of observed distribution. This approach is applicable in situations where other methods cannot be used, e.g. in situations where only moments of the target distribution are available. Some properties of the proposed constructions are derived. In particular, procedures of recovering two types of convolutions, the copula and copula density functions, as well as the conditional density function, are suggested. Finally, the approximation of the inverse Laplace transform is obtained. The performance of moment-recovered construction is illustrated via graphs of a simple density function.     

Estimating copula densities through wavelets

Wavelet analysis is used to construct a rank-based estimator of a copula density. The procedure, which can be easily implemented with ready-to-use wavelet packages, is based on an algorithm that handles boundary effects automatically. The resulting estimator provides a non-parametric benchmark for the selection of a parametric copula family. From a theoretical point of view, the estimation procedure is shown to be optimal in the minimax sense on a large functional class of regular copula densities. The approach is illustrated with actuarial and financial data.     

Adaptive Importance Sampling for simulating copula-based distributions

In this paper, we propose a generalization of importance sampling, called Adaptive Importance Sampling, to approximate simulation of copula-based distributions. Unlike existing methods for copula simulation that have appeared in the literature, this algorithm is broad enough to be used for any absolutely continuous copula. We provide details of the algorithm including rules for stopping the iterative process and consequently assess its performance using extensive Monte Carlo experiments. To assist in its extension to several dimensions, we discuss procedures for identifying the crucial parameters in order to achieve desirable results especially as the size of the dimension increases. Finally, for practical illustration, we demonstrate the use of the algorithm to price First-to-Default credit swap, an important credit derivative instrument in the financial market. The method works exquisitely well even for large dimensions making it a valuable tool for simulating from many different classes of copulas including those which have been difficult to sample from using traditional techniques.     

Flexible Multivariate Density Estimation With Marginal Adaptation

This article is concerned with multivariate density estimation. We discuss deficiencies in two popular multivariate density estimators—mixture and copula estimators, and propose a new class of estimators that combines the advantages of both mixture and copula modeling, while being more robust to their weaknesses. Our method adapts any multivariate density estimator using information obtained by separately estimating the marginals. We propose two marginally adapted estimators based on a multivariate mixture of normals and a mixture of factor analyzers estimators. These estimators are implemented using computationally efficient split-and-elimination variational Bayes algorithms. It is shown through simulation and real-data examples that the marginally adapted estimators are capable of improving on their original estimators and compare favorably with other existing methods. Supplementary materials for this article are available online.     

多元Copula密度估计的小波局部阈值方法

为了量化资产之间相依结构的局部特征,本文将小波阈值规则引入Copula参数估计,提出多元Copula密度的小波局部阈值估计量,发现Copula密度的光滑度指数、维数和采样容量是影响估值精度的重要因素,这一点也得到了以正态Copula为仿真算例的支持。本方法增强了参数Copula建模的局部自适应能力,进而有助于改进资产的市场风险估值与最优化配置。     

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Estimating copula densities, using model selection techniques

Recently a new way of modeling dependence has been introduced considering a sequence of parametric copula models, covering more and more dependency aspects and thus giving a closer approximation to the true copula density. The method uses contamination families based on Legendre polynomials. It has been shown that in general after a few steps accurate approximations are obtained. In this paper selection of the adequate number of steps is considered, and estimation of the unknown parameters within the chosen contamination family is established, thus obtaining an estimator of the unknown copula density. There should be a balance between the complexity of the model and the number of parameters to be estimated. High complexity gives a low model error, but a large stochastic or estimation error, while a very simple model gives a small stochastic error, but a large model error. Techniques from model selection are applied, thus letting the data tell us which aspects are important enough to capture into the model. Natural and simple estimators of the involved Fourier coefficients complete the procedure. Theoretical results show that the expected quadratic error is reduced by the selection rule to the same order of magnitude as in a classical parametric problem. The method is applied on a real data set, illustrating that the new method describes the data set very well: the error involved in the classical Gaussian copula density is reduced with no fewer than 50%.     

Quantifying the risk using copulae with nonparametric marginals

We show that copulae and kernel estimation can be mixed to estimate the risk of an economic loss. We analyze the properties of the Sarmanov copula. We find that the maximum pseudo-likelihood estimation of the dependence parameter associated with the copula with double transformed kernel estimation to estimate marginal cumulative distribution functions is a useful method for approximating the risk of extreme dependent losses when we have large data sets. We use a bivariate sample of losses from a real database of auto insurance claims.     

Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.     

Nonparametric estimation of interval-censored failure time data in the presence of informative censoring

Nonparametric estimation of a survival function is one of the most commonly asked questions in the analysis of failure time data and for this, a number of procedures have been developed under various types of censoring structures (Kalbfleisch and Prentice, 2002). In particular, several algorithms are available for interval-censored failure time data with independent censoring mechanism (Sun, 2006; Turnbull, 1976). In this paper, we consider the interval-censored data where the censoring mechanism may be related to the failure time of interest, for which there does not seem to exist a nonparametric estimation procedure. It is well-known that with informative censoring, the estimation is possible only under some assumptions. To attack the problem, we take a copula model approach to model the relationship between the failure time of interest and censoring variables and present a simple nonparametric estimation procedure. The method allows one to conduct a sensitivity analysis among others.     

Copula函数的选择：方法与应用

针对目前Copula函数在实际应用中的选择问题,本文通过非参数法得到了它们的分布函数图及其经验分布图并进行了比较,然后利用一种解析法对其进一步的选择,并通过Q-Q图比较了各种模型的拟合程度,最后进行了拟合优度检验,得到了最优的Copula。最后对国内的上证A股指数和上证B股指数进行了实证分析,结果体现了该方法的有效性。     

Computing the Volume of n-Dimensional Copulas

Abstract

A problem that is very relevant in applications of copula functions to finance is the computation of the survival copula, which is applied to enforce multivariate put–call parity. This may be very complex for large dimensions. The problem is a special case of the more general problem of volume computation in high-dimensional copulas. We provide an algorithm for the exact computation of the volume of copula functions in cases where the copula function is computable in closed form. We apply the algorithm to the problem of computing the survival of a copula function in the pricing problem of a multivariate digital option, and we provide evidence that this is feasible for baskets of up to 20 underlying assets, with acceptable CPU time performance.     

基于Copula函数的股指尾部相关性研究——以道琼斯工业指数与恒生指数为例

鉴于两步参数估计法在应用中存在误差大、计算复杂等缺陷,采用基于经验分布的半参数估计与非参数估计法确定相应边缘分布与Copula参数,对突发事件下的道琼斯工业指数与恒生指数之间的尾部相关性进行量化.研究发现ClaytonCopula,Gumbel Copula能够较好地刻画股指收益率序列间的尾部相关关系;道指与恒生指数存在着正的尾部相关且这种相关是非对称性的;在各个置信水平上,下尾损失均较上尾收益高,且下尾相关系数的增长幅度远大于上尾相关系数的增长幅度;极端事件造成的道指收益的剧烈下跌引发了恒生指数收益更强烈的相关反应,其造成的影响远超过两个市场同时上涨时的作用.     

The Generalized Cross Entropy Method, with Applications to Probability Density Estimation

Nonparametric density estimation aims to determine the sparsest model that explains a given set of empirical data and which uses as few assumptions as possible. Many of the currently existing methods do not provide a sparse solution to the problem and rely on asymptotic approximations. In this paper we describe a framework for density estimation which uses information-theoretic measures of model complexity with the aim of constructing a sparse density estimator that does not rely on large sample approximations. The effectiveness of the approach is demonstrated through an application to some well-known density estimation test cases.     

Particle Approximations of the Score and Observed Information Matrix for Parameter Estimation in State–Space Models With Linear Computational Cost

Poyiadjis, Doucet, and Singh showed how particle methods can be used to estimate both the score and the observed information matrix for state–space models. These methods either suffer from a computational cost that is quadratic in the number of particles, or produce estimates whose variance increases quadratically with the amount of data. This article introduces an alternative approach for estimating these terms at a computational cost that is linear in the number of particles. The method is derived using a combination of kernel density estimation, to avoid the particle degeneracy that causes the quadratically increasing variance, and Rao–Blackwellization. Crucially, we show the method is robust to the choice of bandwidth within the kernel density estimation, as it has good asymptotic properties regardless of this choice. Our estimates of the score and observed information matrix can be used within both online and batch procedures for estimating parameters for state–space models. Empirical results show improved parameter estimates compared to existing methods at a significantly reduced computational cost. Supplementary materials including code are available.