$L_p$-norm spherical copulas

TL;DR

This paper investigates $L_p$-norm spherical copulas across various dimensions, providing their explicit formulas, conditions for existence, and methods for statistical inference and simulation.

Contribution

It fully characterizes the existence, uniqueness, and explicit formulas of $L_p$-norm spherical copulas for all $p$ and dimensions, advancing understanding of their properties.

Findings

Explicit formulas for densities and correlation coefficients.

Conditions for existence and uniqueness established.

Distribution of the radial part determined.

Abstract

In this paper we study $L_p$-norm spherical copulas for arbitrary $p \in [1,\infty]$ and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean difference. We fully characterize conditions for existence and uniqueness of $L_p$-norm spherical copulas. Explicit formulas for their densities and correlation coefficients are derived and the distribution of the radial part is determined. Moreover, statistical inference and efficient simulation are considered.