# Dependence Properties of B-Spline Copulas

## TL;DR

This paper introduces a new class of copulas constructed with B-spline functions, extending classical Bernstein copulas, and explores their correlation properties, bounds, and moments with explicit formulas.

## Contribution

It develops B-spline copulas, generalizing Bernstein copulas, and proves their correlation bounds and total positivity properties, including explicit moment formulas using Stirling numbers.

## Key findings

- Range of correlation is characterized.
- Frechet--Hoeffding upper bound is attained with infinite B-spline functions.
- Explicit moments are derived using Stirling numbers.

## Abstract

We construct by using B-spline functions a class of copulas that includes the Bernstein copulas arising in Baker's distributions. The range of correlation of the B-spline copulas is examined, and the Frechet--Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system from which the property of total positivity of any order follows for the maximum correlation case, the results given here extend classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions for nonnegative real numbers.