STATISTICAL-INFERENCE PROCEDURES FOR BIVARIATE ARCHIMEDEAN COPULAS
STATISTICAL-INFERENCE PROCEDURES FOR BIVARIATE ARCHIMEDEAN COPULAS
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DOI：
10.2307/2290796
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发表时间：
1993-09-01
影响因子：
3.7
通讯作者：
RIVEST, LP
中科院分区：
文献类型：
--
作者：
GENEST, C;RIVEST, LP
一个具有边值F（x）和G（y）的二元分布函数H（x，y）被称为由阿基米德copula生成的，如果它可以表示为H（x，y）= phi-1[phi{F（x）} + phi{G（y）}]的形式，对于定义在（0，1]上的某个凸的递减函数phi，以这样的方式，phi（1）= 0。许多著名的二元分布系统都属于这一类，包括Gumbel、Ali-Mikhail-Haq-Thelot、克莱顿、Frank和Hougaard的系统。脆弱的模型也属于一般处方。本文研究了在随机样本（X1，Y1），.. (X(n)，Y（n））。估计过程的关键是一个一维的经验分布函数，可以构造X和Y的统一表示是否是阿基米德的，并且独立于它们的边缘。这个半参数估计，基于分解的肯德尔的tau统计量，被认为是平方根n-一致的，并提供了一个明确的公式，其渐近方差。这导致了一个战略选择参数家庭的阿基米德copula，提供了最好的可能适合一组给定的数据。为了说明这些程序，铀矿勘探数据集重新分析。虽然介绍仅限于涉及从一个二元分布的随机样本的问题，扩展到涉及多变量或删失数据的情况下，可以设想。
A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = phi-1[phi{F(x)} + phi{G(y)}] for some convex, decreasing function phi defined on (0, 1] in such a way that phi(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X1, Y1),..., (X(n), Y(n)). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be square-root n-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems involving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged.