New estimators of the Pickands dependence function and a test for extreme-value dependence
Pickands 依赖函数的新估计量与极值依赖检验AI 解读 ↓
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Pickands 依赖函数的新估计量与极值依赖检验
- 研究问题
- 如何估计 Pickands 依赖函数，并利用估计过程检验 copula 是否具有极值依赖，尤其如何处理非极值备择？
- 方法与思路
- 作者以对数 copula 的加权平方距离构造最小距离目标，再以经验 copula 替换未知对象，研究弱收敛并构造检验。
- 材料陈述的贡献
- 摘要称部分新估计量在理论及模拟中表现更好，且所得检验对满足一阶弱可微条件的正象限依赖备择一致。
- 阅读关注点
- 核对权重选择、目标函数的边界与形状性质、弱收敛条件，以及效率比较标准和检验一致性的备择范围。
- 阅读边界
- 版本未注明的索引摘要未给出具体权重与比较数值；满足边界条件不等于满足 Pickands 函数的全部要求。
原文摘要
We propose a new class of estimators for Pickands dependence function which is based on the concept of minimum distance estimation. An explicit integral representation of the function A*(t), which minimizes a weighted L2-distance between the logarithm of the copula C(y1−t, yt) and functions of the form A(t)log(y) is derived. If the unknown copula is an extreme-value copula, the function A*(t) coincides with Pickands dependence function. Moreover, even if this is not the case, the function A*(t) always satisfies the boundary conditions of a Pickands dependence function. The estimators are obtained by replacing the unknown copula by its empirical counterpart and weak convergence of the corresponding process is shown. A comparison with the commonly used estimators is performed from a theoretical point of view and by means of a simulation study. Our asymptotic and numerical results indicate that some of the new estimators outperform the estimators, which were recently proposed by Genest and Segers [Ann. Statist. 37 (2009) 2990–3022]. As a by-product of our results, we obtain a simple test for the hypothesis of an extreme-value copula, which is consistent against all positive quadrant dependent alternatives satisfying weak differentiability assumptions of first order.