论文标题
EFRON定理的概括
Generalizations of Efron's theorem
论文作者
论文摘要
在本文中，我们证明了[EFR65]中Efron证明的两个定理的新版本。 Efron的定理说，如果函数$ ϕ：\ mathbb {r}^2 \ rightArrow \ Mathbb {r} $在每个参数中都不予以解决，那么我们就有函数$ s \ mapsto \ mapsto \ mathbb {e} e} [E} [E}]我们将限制的efron定理命名为efron定理的版本，其中$ ϕ：\ mathbb {r} \ rightarrow \ mathbb {r} $仅取决于一个变量。 $ pf_n $是诸如$ \ forall a_1 \ leq ... \ leq a_n，b_1 \ leq ... \ leq ... \ leq b_n，\ det（f（a_i-b_j））_ {1 {1 \ leq i，j \ leq n} \ geq 0的$ pranial forem forem forfron，$ pf_n $是$ \ forall a_1 \ leq ... 班级。第二个以更强的单调性假设来考虑非限制的EFRON定理。在最后一部分中，我们给出了EFRON定理的第二个概括的更一般结果。
In this article, we prove two new versions of a theorem proven by Efron in [Efr65]. Efron's theorem says that if a function $ϕ: \mathbb{R}^2 \rightarrow \mathbb{R}$ is non-decreasing in each argument then we have that the function $s \mapsto \mathbb{E}[ϕ(X,Y)|X+Y=s]$ is non-decreasing. We name restricted Efron's theorem a version of Efron's theorem where $ϕ: \mathbb{R} \rightarrow \mathbb{R}$ only depends on one variable. $PF_n$ is the class of functions such as $\forall a_1 \leq ... \leq a_n, b_1 \leq ... \leq b_n, \det(f(a_i-b_j))_{1 \leq i,j \leq n} \geq 0.$ The first version generalizes the restricted Efron's theorem for random variables in the $PF_n$ class. The second one considers the non-restricted Efron's theorem with a stronger monotonicity assumption. In the last part, we give a more general result of the second generalization of Efron's theorem.
