Abstract

This article explores the parameter space of multivariate skew normal families having identical distributed marginal distributions under a few autoregressive-moving average correlation structures ($\overline{\mathbf{\Omega }}$): namely, MA(1), MA(2), and AR(2) correlation structures. Such an undertaking escapes triviality since the efficacious $\{\overline{\mathbf{\Omega }},\mathit{\delta }\}$ parametrization, in terms of analysis of marginal distributions, restricts the parameter space of the correlation and shape parameters in an intertwined fashion, which upon unraveling illuminates properties of the multivariate skew normal’s correlation matrix, where under identical marginals, $\mathit{\delta }=\delta \mathbf{1}.$ Using the $\{\overline{\mathbf{\Omega }},\mathit{\delta }\}$ parametrization of the multivariate skew normal distribution, the support of δ is found using a limiting argument for the cases considered herein, which can be approximated well via nonlinear regression using a ratio of rational functions of the sample size, k, for an MA(q) correlation structure. Moreover, a range of values for δ is found, when possible, such that the parameter space of the correlation parameters under skew-normality agrees with that under normality. For fixed δ’s, the parameter space of the correlation parameters is found by numerical inversion, where plots have been included to visualize the regions. From the parameter space, inequalities on the component-wise correlations of the skew normal families are developed and displayed pictorially. Finally, transformations are offered to convert the multivariate skew normal families to the more prevalent $\{\overline{\mathbf{\Omega }},\mathit{\alpha }\}$ parametrization, with $\mathit{\alpha }$ representing a shape parameter.

摘要

本文探讨了在几个自回归移动平均相关结构下具有相同分布边际分布的多元偏斜正态族的参数空间（$\overline{\mathbf{欧姆}}$): 即 MA(1)、MA(2) 和 AR(2) 相关结构。由于有效$\{\overline{\mathbf{欧姆}},\mathit{\delta }\}$参数化，就边际分布分析而言，以交织的方式限制了相关性和形状参数的参数空间，这在解开时阐明了多元偏斜法线相关矩阵的特性，其中在相同的边际下，$\mathit{\delta }=\delta \mathbf{1个}.$使用$\{\overline{\mathbf{欧姆}},\mathit{\delta }\}$多元偏斜正态分布的参数化，使用此处考虑的情况的限制参数找到δ的支持，这可以通过使用样本大小的有理函数比率k的非线性回归很好地近似，对于 MA( q ) 相关结构。此外，在可能的情况下，找到δ的值范围，使得相关参数的参数空间在偏态正态性下与正态性下的参数空间一致。对于固定的δ的，相关参数的参数空间是通过数值反演找到的，其中包含了绘图以可视化区域。从参数空间，偏态正态族的分量相关性的不等式被开发并以图形方式显示。最后，提供转换以将多变量偏态正常家庭转换为更普遍的$\{\overline{\mathbf{欧姆}},\mathit{\alpha }\}$参数化，与$\mathit{\alpha }$表示形状参数。