AI 中文总结
研究二元copula类上肯德尔tau、斯皮尔曼步长规则和布洛姆奎斯特贝塔联合值的精确区域，通过构造性方法确定区域边界及相关性质，包括凸性、对称性等，还计算出区域体积为$\frac{31}{40}$。
AI 中文摘要
我们确定了在所有二元copula类$\mathcal{C}$上，肯德尔tau、斯皮尔曼步长规则和布洛姆奎斯特贝塔可能的联合值的精确区域$\Omega_{\tau,\phi,\beta}:=\{(\tau(C),\phi(C),\beta(C)):C\in\mathcal{C}\}$。该区域恰好由所有满足$-1\le b\le 1$，$\frac{3}{16}(1+b)^2-\frac12\le p\le 1-\frac38(1-b)^2$以及$\frac43 p-\frac13\le t\le \frac23 p+\frac13$的三元组$(t,p,b)$组成。证明是构造性的，通过特定的洗牌族和序数和来实现，还表明该区域是凸的且有矩形固定步长规则截面，确定了其纤维关于$\tau=\phi$的仿射对称性并计算出其体积为$\frac{31}{40}$。
英文摘要
We determine the exact region $Ω_{τ,ϕ,β}:=\{(τ(C),ϕ(C),β(C)):C\in\mathcal{C}\}$ of possible joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over the class $\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(t,p,b)$ satisfying $-1\le b\le 1$, $\frac{3}{16}(1+b)^2-\frac12\le p\le 1-\frac38(1-b)^2$ and $\frac43 p-\frac13\le t\le \frac23 p+\frac13$. In other words, the known exact $(ϕ,β)$- and $(τ,ϕ)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional sharp restriction on the possible values of Kendall's tau. The proof is constructive: two one-parameter families of shuffles of $M$ realize the extreme values of Kendall's tau along the lower boundary of the $(ϕ,β)$-region, ordinal sums spread these families through the whole region, and the vertical fibres are filled using the biaffinity of the concordance function. We further show that $Ω_{τ,ϕ,β}$ is convex with rectangular fixed-footrule sections, identify an affine symmetry of its fibres about $τ=ϕ$, and compute its volume, which equals $\frac{31}{40}$.
