Some properties of skew-symmetric distributions

Abstract

The family of skew-symmetric distributions is a wide set of probability density functions obtained by suitably combining a few components which can be quite freely selected provided some simple requirements are satisfied. Although intense recent work has produced several results for certain sub-families of this construction, much less is known in general terms. The present paper explores some questions within this framework, and provides conditions for the above-mentioned components to, ensure that the final distribution enjoys specific properties

Stochastic ordering of medians in exchangeable trivariate normal vectors

“…Many of the properties of the SN 2 distribution in (9) are similar to those of the SN distribution in (6). For example, it reduces to the standard normal distribution when λ = 0 and it is strongly unimodal for all values of λ ∈ R (strong unimodality can be easily obtained by the results in Azzalini and Regoli [2] and Loperfido et al [7]). Moreover, just like the SN distribution, the SN 2 distribution in (9) is stochastically ordered with respect to λ.…”

“…Let X = (X 1 , · · · , X n ) T and Y = (Y 1 , · · · , Y n ) T be two n-dimensional exchangeable normal random vectors as X ∼ N n μ1 n , σ 2 (1 − ρ X ) I n + ρ X 1 n 1 T n , Y ∼ N n μ1 n , σ 2 …”

Bayesian Analysis of Regression Models Using Instrumental Variables: A Case Study (Iranian Rural Households Income and Expenditure Data)

“…Several generalizations of these were presented by Balakrishnan (2002), Genton (2004), Gupta et al (2004) and Arellano-valle et al (2010). Recently, Azzalini and Regoli (2012) investigated other interesting properties of skew-symmetric distribution.…”

A general setting for symmetric distributions and their relationship to general distributions

“…where equality of the inner integrals in the last two lines follows from (10). Thus the measure µ is decomposed (disintegrated) in the sense of [12,Section 5] or [14] into the marginal measure π * µ on X /K and the conditional measures ν [x] on the fibres π −1 ([x]).…”

“…The construction is described in Corollary 1 below and is a direct extension of that for s = 2 appearing at the end of Section 2.1 in [11], which in turn is related to expression (9) of [10]; all these formulations are generalisations of representation (4) for the case K = C 2 . Corollary 1.…”