Generalizations of Efron's theorem

In this article, we prove two new versions of a theorem proven by Efron in [Efr65]. Efron's theorem says that if a function $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ is non-decreasing in each argument then we have that the function $s \mapsto \mathbb{E}[\phi(X,Y)|X+Y=s]$ is non-decreasing. We name restricted Efron's theorem a version of Efron's theorem where $\phi : \mathbb{R} \rightarrow \mathbb{R}$ 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.