The multivariate skew normal (MSN) and multivariate skew t (MST) distributions have received considerable attention in the past two decades because of their appealing mathematical properties and their usefulness for modeling skewed data. We develop sparse regularization methodology for estimating the skewness parameters of these two distributions. This methodology facilitates skewness selection, i.e., the identification of those marginal indices of skewness, if any, that are equal to zero. Obstacles that render skewness selection infeasible for existing parameterizations of the two distributions are described, and a new parameterization that permits the circumvention of those obstacles is introduced. A penalized likelihood method for sparsity-based regularized skewness estimation using the new parameterization is proposed. Model selection consistency and the oracle property of the method are established. A simulation study demonstrates that the method is reasonably effective for skewness selection and, because of inclusion of a ridge penalty, is more effective at preventing the divergence of the shape parameter in small samples than the Q-penalty approach of Azzalini and Arellano-Valle (2013). The simulation study demonstrates further that our method may improve the estimation of all parameters of the MSN and MST distributions, not merely the skewnesses, when some of the skewnesses are zero.

中文翻译：

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多元偏态正态分布和多元偏态 t 分布的稀疏正则化偏度估计

在过去的二十年中，多元偏斜正态 (MSN) 和多元偏斜 t (MST) 分布受到了相当多的关注，因为它们具有吸引人的数学特性以及它们对偏斜数据建模的有用性。我们开发了稀疏正则化方法来估计这两个分布的偏度参数。这种方法有助于偏度选择，即识别那些等于零的偏度边缘指数（如果有的话）。描述了使偏度选择对于两个分布的现有参数化不可行的障碍，并引入了允许规避这些障碍的新参数化。提出了一种使用新参数化的基于稀疏性的正则化偏度估计的惩罚似然方法。建立了模型选择的一致性和方法的预言性。模拟研究表明，该方法对于偏度选择相当有效，并且由于包含脊惩罚，在防止小样本中形状参数发散方面比 Azzalini 和 Arellano-Valle 的 Q 惩罚方法更有效（ 2013）。模拟研究进一步表明，当一些偏度为零时，我们的方法可以改进对 MSN 和 MST 分布的所有参数的估计，而不仅仅是偏度。与 Azzalini 和 Arellano-Valle (2013) 的 Q 惩罚方法相比，在防止小样本中形状参数发散方面更有效。模拟研究进一步表明，当一些偏度为零时，我们的方法可以改进对 MSN 和 MST 分布的所有参数的估计，而不仅仅是偏度。与 Azzalini 和 Arellano-Valle (2013) 的 Q 惩罚方法相比，在防止小样本中形状参数发散方面更有效。模拟研究进一步表明，当一些偏度为零时，我们的方法可以改进对 MSN 和 MST 分布的所有参数的估计，而不仅仅是偏度。