On the problem of the maximal asymmetry projection in the family of scale mixtures of skew-normal vectors
The class of multivariate scale mixtures of skew-normal (SMSN) distributions is a flexible family that accounts for the non-normality of the data by means of a tail weight parameter and a shape vector that regulates the asymmetry of the model. Its stochastic representation involves a skew-normal (SN) vector and a non-negative mixing scalar variable, independent of the SN vector, that injects tail weight into the multivariate model. We address the problem of finding the direction yielding the projection with the maximal asymmetry for SMSN vectors. It can be shown that such direction is proportional to the shape vector when a simple condition on the moments of the mixing variable is fulfilled. This finding highlights the directional nature of the shape vector to regulate the multivariate asymmetry for this class of distributions; it also contributes to the theoretical foundations of the skewness model based projection pursuit problem in the SMSN family.
Palabras clave / Keywords: skew-normal scale mixture of skew-normal distributions maximal skewness moments mixing variable
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