Complete asymptotic expansions and the high-dimensional Bingham distributions

Abstract

For \(d \ge 2\), let X be a random vector having a Bingham distribution on \({\mathcal {S}}^{d-1}\), the unit sphere centered at the origin in \({\mathbb {R}}^d\), and let \(\Sigma \) denote the symmetric matrix parameter of the distribution. Let \(\Psi (\Sigma )\) be the normalizing constant of the distribution and let \(\nabla \Psi _d(\Sigma )\) be the matrix of first-order partial derivatives of \(\Psi (\Sigma )\) with respect to the entries of \(\Sigma \). We derive complete asymptotic expansions for \(\Psi (\Sigma )\) and \(\nabla \Psi _d(\Sigma )\), as \(d \rightarrow \infty \); these expansions are obtained subject to the growth condition that \(\Vert \Sigma \Vert \), the Frobenius norm of \(\Sigma \), satisfies \(\Vert \Sigma \Vert \le \gamma _0 d^{r/2}\) for all d, where \(\gamma _0 > 0\) and \(r \in [0,1)\). Consequently, we obtain for the covariance matrix of X an asymptotic expansion up to terms of arbitrary degree in \(\Sigma \). Using a range of values of d that have appeared in a variety of applications of high-dimensional spherical data analysis, we tabulate the bounds on the remainder terms in the expansions of \(\Psi (\Sigma )\) and \(\nabla \Psi _d(\Sigma )\) and we demonstrate the rapid convergence of the bounds to zero as r decreases.