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Modeling and Inference with v-Spherical Distributions

Carmen Fernández, Jacek Osiewalski and Mark F. J. Steel
Journal of the American Statistical Association
Vol. 90, No. 432 (Dec., 1995), pp. 1331-1340
DOI: 10.2307/2291523
Stable URL: http://www.jstor.org/stable/2291523
Page Count: 10
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Modeling and Inference with v-Spherical Distributions
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Abstract

A new class of continuous multivariate distributions on X ∈ Rn is proposed. We define these so-called ν-spherical distributions through properties of the density function in a location-scale context. We derive conditions for properness of ν-spherical distributions and discuss how to generate them in practice. The name "ν-spherical" is motivated by the fact that these distributions generalize the classes of spherical (when ν(·) is the l2 norm) and lq-spherical (when ν(·) is the lq 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 lq-spherical models carry over directly to ν-spherical models. In particular, it is shown that under a common improper prior on the scale parameter τ-1, any ν-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.

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