$L_p$-norm spherical copulas

In this paper we study $L_p$-norm spherical copulas for arbitrary $p\in [1,\mathrm{\infty }]$ 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.