Modeling and Inference with υ-Spherical Distributions

Abstract

A new class of continuous multivariate distributions on × ∈ ℜ ⁿ is proposed. We define these so-called υ-spherical distributions through properties of the density function in a location-scale context. We derive conditions for properness of υ-spherical distributions and discuss how to generate them in practice. The name “υ-spherical” is motivated by the fact that these distributions generalize the classes of spherical (when υ(·) is the l ₂ norm) and l _q -spherical (when υ(·) is the l _q norm) distributions. Isodensity sets are still always situated around the location parameter μ, but exchangeability and axial symmetry are no longer imposed, as is illustrated in some examples. As an important special case, we define a class of distributions suggested by independent sampling from a generalization of exponential power distributions. This allows us to model skewness. Interestingly, all the robustness results found previously for spherical and l _q -spherical models carry over directly to υ-spherical models. In particular, it is shown that under a common improper prior on the scale parameter τ⁻¹, any υ-spherical distribution with the same isodensity sets will lead to the same density p(x, μ). Under proper priors on τ, we can still find some robustness results, although of lesser generality.