Functions operating on several multivariate distribution functions

Abstract

Functions ff on [0,1]m{\left[0,1]}^{m} such that every composition f∘(g1,…,gm)f\circ \left({g}_{1},\ldots ,{g}_{m}) with dd-dimensional distribution functions g1,…,gm{g}_{1},\ldots ,{g}_{m} is again a distribution function, turn out to be characterized by a very natural monotonicity condition, which for d=2d=2 means ultramodularity. For m=1m=1 (and d=2d=2), this is equivalent with increasing convexity.

Keywords

• multivariate distribution function
• ultramodular
• bernstein polynomials
• faà di bruno’s formula
• higher order
• monotonicity
• 26e05
• 26b40
• 60e05