Generalizations of Efron's theorem

In this article, we prove two new versions of a theorem proven by Efron in Efron (1965). Efron's theorem says that if a function phi : R-2 -> R is non-decreasing in each argument then we have that the function s bar right arrow E[phi(X, Y)vertical bar X + Y = s] is non decreasing. We name restricted Efron's theorem a version of Efron's theorem where phi : R -> R only depends on one variable. PFn is the class of functions such as for all a(1) <= ... <= a(n), b(1) <= ... <= b(n), det(f (a(i) - b(j)))(1 <= i,j <= n) >= 0. The first version generalizes the restricted Efron's theorem for random variables in the PFn 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. (C) 2021 Elsevier B.V. All rights reserved.