Reference Work Entry

International Encyclopedia of Statistical Science

pp 1344-1346


Skew-Symmetric Families of Distributions

  • Adelchi AzzaliniAffiliated withUniversity of Padua

The term ‘skew-symmetric distributions’ refers to the construction of a continuous probability distribution obtained by applying a certain form of perturbation to a symmetric density function.

To be more specific, a concept of symmetric distribution must be adopted first, since in the multivariate setting various forms of symmetry have been introduced. The variant used in this context is the one of central symmetry, a natural extension of the traditional one-dimensional form to the d-dimensional case: if f0 is a density function on ℝd and ξ is a point of ℝd, central symmetry around ξ requires that f0(t − ξ) = f0( − t − ξ) for all t ∈ ℝd, ignoring sets of 0 probability. To avoid notational complications, we shall concentrate on the case with ξ = 0; it is immediate to rephrase what follows in the case of general ξ, which simply amounts to a shift of the location of the distribution.

If f0 is a probability density function on ℝd centrally symmetric around 0, there are two largely equivalent ...

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