On the exact region determined by Kendall's τ and Spearman's ρ

• 成果类型: Article
• 署名作者: Schreyer, Manuela; Paulin, Roland; Trutschnig, Wolfgang
• 署名单位: Salzburg University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• 发表日期: 2017
• 页码: 613-633
• 关键词: copulas

摘要

Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region determined by all possible values of Kendall's and Spearman's . In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is not sharp outside a countable set, give a simple analytic characterization of in terms of a continuous, strictly increasing piecewise concave function and show that is compact and simply connected, but not convex. The results also show that for each (x,y) epsilon Omega there are mutually completely dependent random variables X and Y whose tau- and rho-values coincide with x and y respectively.