Simplified matrix methods are used to analyze the higher order asymptotic properties of \(k\times 1\) sample averages. Kronecker differentiation is used to define \(k^{j }\times 1\), j ’th order moments, \(\mu _j\), cumulants \(\kappa _j\) and Hermite polynomials, \(H_j\). These are then used to derive valid multivariate Edgeworth expansions of arbitrary order having the same form as the standard univariate case: \(p(x) = \phi (x)[1 + N^{-1/2} \kappa _{3}' H_{ 3}(x) /6 +{ N^{-1} } ( 3 { \kappa _{4}' }{ } H_{ 4}(x) + \kappa _3'^{ \otimes 2} H_{ 6}(x) )/72+\cdots ]\). All the usual steps in the development of a valid Edgeworth expansion are shown to be easily derived using matrix algebra.

中文翻译：

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多元Edgeworth展开的简化矩阵方法

简化的矩阵方法用于分析\（k \ times 1 \）样本平均值的高阶渐近性质。Kronecker微分用于定义\（k ^ {j} \ times 1 \），第 j 个阶矩，\（\ mu _j \），累积量\（\ kappa _j \）和Hermite多项式\（H_j \） 。然后将它们用于导出任意阶的有效多元Edgeworth展开，其形式与标准单变量情况相同：\（p（x）= \ phi（x）[1 + N ^ {-1/2} \ kappa _ { 3}'H_ {3}（x）/ 6 + {N ^ {-1}}（3 {\ kappa _ {4}'} {} H_ {4}（x）+ \ kappa _3'^ {\ otimes 2} H_ {6}（x））/ 72 + \ cdots] \）。有效的Edgeworth扩展开发过程中的所有常规步骤都证明可以使用矩阵代数轻松推导。