Simplified Matrix Methods for Multivariate Edgeworth Expansions

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