5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in n − 1 / 2 about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate w ^ of an unknown vector w in R p , as a standard estimate , if E w ^ → w as n → ∞ , and for r ≥ 1 the r th-order cumulants of w ^ have magnitude n 1 − r and can be expanded in n − 1 . Here we present a significant extension. We give the expansion of the distribution of any smooth function of w ^ , say t ( w ^ ) in R q , giving its distribution to n − 5 / 2 . We do this by showing that t ( w ^ ) , is a standard estimate of t ( w ) . This provides far more accurate approximations for the distribution of t ( w ^ ) than its asymptotic normality.