Multivariate scale mixtures of skew-normal distributions are flexible models that account for the non-normality of data by means of a tail weight parameter and a shape vector representing the asymmetry of the model in a directional fashion. Its stochastic representation involves a skew-normal vector and a non negative mixing scalar variable, independent of the skew-normal vector, that injects tail weight behavior into the model. In this paper we look into the problem of finding the projection that maximizes skewness for vectors that follow a scale mixture of skew-normal distribution; when a simple condition on the moments of the mixing variable is fulfilled, it can be shown that the direction yielding the maximal skewness is proportional to the shape vector. This finding stresses the directional nature of the shape vector to regulate the asymmetry; it also provides the theoretical foundations motivating the skewness based projection pursuit problem in this class of distributions. Some examples that illustrate the application of our results are also given; they include a simulation experiment with artificial data, which sheds light on the usefulness and implications of our results, and the application to real data.

中文翻译：

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在偏正态向量的比例混合下通过偏度最大化进行数据投影

偏态正态分布的多元尺度混合是灵活的模型，它通过尾部权重参数和以方向性方式表示模型不对称性的形状矢量来说明数据的非正态性。它的随机表示涉及一个偏态法线向量和一个非负混合标量变量，与偏态法线向量无关，该变量将尾部重量行为注入模型。在本文中，我们研究了以下问题：找到遵循正态正态分布比例混合的向量的最大化偏度的投影；当满足混合变量矩的简单条件时，可以证明产生最大偏斜的方向与形状矢量成比例。这一发现强调了形状矢量的方向性以调节不对称性。它也为激发此类分布中基于偏度的投影追踪问题提供了理论基础。还给出了一些例子来说明我们的结果的应用。其中包括使用人工数据进行的模拟实验，阐明了我们的结果的有用性和含义，以及对实际数据的应用。