By analogy to the real case established by Matusita (1955) we introduce the concept of affinity between two complex distribution functions. We also establish a concrete expression for the affinity between two complex k-variate normal distributions when the covariance matrices assume a special form. Generalizations of these results are presented and the expressions here obtained are compared with those obtained by Matusita (1966, 1967) relative to the affinity between real k-variate normal distributions....

We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

Let (X,Y) be a random vector with joint probability measure σ and with margins μ and ν. Let ${\left(Pₙ\right)}_{n\in \mathbb{N}}$ and ${\left(Qₙ\right)}_{n\in \mathbb{N}}$ be two bases of complete orthonormal polynomials with respect to μ and ν, respectively. Under integrability conditions we have the following polynomial expansion: $\sigma (dx,dy)={\sum }_{n,k\in \mathbb{N}}{\varrho }_{n,k}Pₙ\left(x\right)Q_k\left(y\right)\mu \left(dx\right)\nu \left(dy\right)$. In this paper we consider the problem of changing the margin μ into μ̃ in this expansion. That is the case when μ is the true (or estimated) margin and μ̃ is its approximation. It is shown that a new joint probability with new margins...

We complement the recently introduced classes of lower and upper semilinear copulas by two new classes, called vertical and horizontal semilinear copulas, and characterize the corresponding class of diagonals. The new copulas are in essence asymmetric, with maximum asymmetry given by $1/16$. The only symmetric members turn out to be also lower and upper semilinear copulas, namely convex sums of $\Pi$ and $M$.